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| − | == Catalog Information ==
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| − |
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| − | === Title ===
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| − | Mathematics for Engineering 1.
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| − |
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| − | === (Credit Hours:Lecture Hours:Lab Hours) ===
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| − | (4:4:0)
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| − |
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| − | === Offered ===
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| − | F, W
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| − |
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| − | === Prerequisite ===
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| − | [[Math 113]] and passing grade on required preparatory exam taken during first week of class. (Practice exams available on class website.)
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| − |
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| − | === Description ===
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| − | Multivariable calculus, linear algebra, and numerical methods.
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| − |
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| − | == Desired Learning Outcomes ==
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| − | This course is designed to give students from the College of Engineering and Technology the mathematics background necessary to succeed in their chosen field.
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| − |
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| − | === Prerequisites ===
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| − |
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| − | Students are expected to have completed [[Math 113]].
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| − |
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| − | === Minimal learning outcomes ===
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| − |
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| − | <div style="-moz-column-count:2; column-count:2;">
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| − |
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| − | # Rectangular Space Coordinates; Vectors in Three-Dimensional Space
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| − | #* Define the following:
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| − | #** Cartesian coordinates of a point
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| − | #** sphere
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| − | #** symmetry about a point, a line, and a plane
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| − | #** vector
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| − | #** components of a vector
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| − | #** vector addition
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| − | #** scalar multiplication
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| − | #** zero vector
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| − | #** vector subtraction
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| − | #** vector norm (magnitude, length)
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| − | #** unit vector
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| − | #** coordinate unit vectors i, j, k
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| − | #** linear combination of unit vectors
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| − | #* Plot points in three-dimensional space.
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| − | #* Calculate the distance between two points in two-dimensional space and 3-dimensional space
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| − | #* Write the equation of a sphere centered about a given point with a given radius. Determine the center and radius of a sphere, given its equation.
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| − | #* Write the component equations of a line that passes through two given points.
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| − | #* Write the component equations of a line segment with given endpoints.
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| − | #* Find the midpoint of a given line segment.
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| − | #* Find the points of symmetry about a point, line, or plane.
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| − | #* Represent a vector by each of the following:
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| − | #** components
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| − | #** a linear combination of coordinate unit vectors
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| − | #* Carry out the vector operations:
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| − | #** addition
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| − | #** scalar multiplication
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| − | #** subtraction
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| − | #* Represent the operations of vector addition, scalar multiplication and norm geometrically.
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| − | #* Find the norm (magnitude, length) of a vector. Determine whether two vectors are parallel.
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| − | #* Recall, apply and verify the basic properties of vector addition, scalar multiplication and norm.
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| − | #* Model and solve application problems using vectors.
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| − | # The Dot Product
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| − | #* Define the following:
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| − | #** dot product.
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| − | #** perpendicular vectors.
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| − | #** unit vector in the direction of a vector a, denoted u_a.
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| − | #** the projection of a on b, denoted proj_b a.
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| − | #** the b-component of a, denoted comp_b a.
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| − | #** the direction cosines of a vector.
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| − | #** the direction angles of a vector.
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| − | #** the Schwarz Inequality.
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| − | #** the work done by a constant force on an object.
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| − | #** the dot product test for perpendicular vectors.
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| − | #** the dot product test for parallel vectors.
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| − | #** geometric interpretation of the dot product
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| − | #* Evaluate a dot product from the coordinate formula or the angle formula.
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| − | #* Interpret the dot product geometrically.
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| − | #* Evaluate the following using the dot product:
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| − | #** the length of a vector.
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| − | #** the angle between two vectors.
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| − | #** u_a, the unit vector in the direction of a vector a.
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| − | #** proj_b a, the projection of a on b.
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| − | #** comp_b a, the b-component of a.
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| − | #** the direction cosines of a vector.
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| − | #** the direction angles of a vector.
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| − | #** the work done by a constant force on an object.
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| − | #* Prove and verify the Schwarz Inequality.
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| − | #* Prove and apply the dot product tests for perpendicular and parallel vectors.
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| − | #* Recall and apply the properties of the dot product.
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| − | #* Prove identities involving the dot product.
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| − | #* Solve application problems involving the dot product.
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| − | #* Extend the vector operations and related identities for addition, scalar multiplication, and dot product to higher dimensions.
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| − | # The Cross Product
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| − | #* Define the following:
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| − | #** the cross product of two vectors
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| − | #** scalar triple product
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| − | #* Evaluate a cross product from the the coordinate formula or angle formula.
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| − | #* Interpret the cross product geometrically.
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| − | #* Evaluate the following using the cross product:
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| − | #** a vector perpendicular to two given vectors.
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| − | #** the area of a parallelogram.
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| − | #** the area or a triangle.
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| − | #** moment of force or moment of torque.
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| − | #* Evaluate scalar triple products.
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| − | #* Use the scalar triple product to determine the following:
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| − | #** volume of a parallelepiped.
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| − | #** whether or not three vectors are coplanar.
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| − | #* Recall and apply the properties of the cross product and scalar triple product.
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| − | #* Prove identities involving the cross product and the scalar triple product.
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| − | #* Solve application problems involving the cross product and scalar triple product.
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| − | # Lines
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| − | #*Define the following:
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| − | #** direction vector for a line
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| − | #** vector equation of a line
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| − | #** scalar parametric equations of a line
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| − | #** Cartesian equations or symmetric form of a line
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| − | #* Represent a line in 3-space by:
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| − | #** a vector equation
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| − | #** scalar parametric equations
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| − | #** Cartesian equations
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| − | #* Find the equation(s) representing a line given information about
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| − | #** a point of the line and the direction of the line or
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| − | #** two points contained in the line.
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| − | #** a point and a parallel line.
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| − | #** a point and perpendicular to a plane.
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| − | #** two planes intersecting in the line.
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| − | #* Find the distance from a point to a line.
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| − | #* Solve application problems involving lines.
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| − | # Planes
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| − | #* Define the following:
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| − | #** normal vector to a plane
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| − | #** cartesian equation of a plane
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| − | #** parametric equation of a plane
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| − | #* Find the equation of a plane in 3-space given a point and a normal vector, three points, or a geometric description of the plane.
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| − | #* Determine a normal vector and the intercepts of a given plane.
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| − | #* Represent a plane by parametric equations.
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| − | #* Find the distance from a point to a plane.
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| − | #* Find the angle between a line and a plane.
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| − | #* Determine a point of intersection between a line and a surface.
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| − | #* Sketch planes given their equations.
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| − | #* Solve application problems involving planes.
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| − | # Systems of Linear Equations
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| − | #* Define the following:
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| − | #** linear system of m equations in n unknowns
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| − | #** consistent and inconsistent
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| − | #** solution set
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| − | #** coefficient matrix
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| − | #** elementary row operations
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| − | #* Identify linear systems.
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| − | #* Represent a system of linear equations as an augmented matrix and vice versa.
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| − | #* Relate the following types of solution sets of a system of two or three variables to the intersections of lines in a plane or the intersection of planes in three space:
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| − | #** a unique solution.
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| − | #** infinitely many solutions.
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| − | #** no solution.
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| − | # Gaussian elimination
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| − | #* Define the following:
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| − | #** reduced row echelon form
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| − | #** leading variables or pivots
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| − | #** free variables
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| − | #** row echelon form
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| − | #** back substitution
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| − | #** Gaussian elimination
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| − | #** Gauss-Jordan elimination
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| − | #** homogeneous
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| − | #** trivial solution
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| − | #** nontrivial solutions
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| − | #* Identify matrices that are in row echelon form and reduced row echelon form.
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| − | #* Determine whether a linear system is consistent or inconsistent from its reduced row echelon form. If the system is consistent, write the solution.
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| − | #* Identify the lead variables and free variables of a system represented by an augmented matrix in reduced row echelon form.
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| − | #* Solve systems of linear equations using Gaussian elimination and back substitution.
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| − | #* Solve systems of linear equations using Gauss-Jordan elimination.
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| − | #* Model and solve application problems using linear systems.
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| − | # Matrices and Matrix Operations
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| − | #* Define the following:
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| − | #** vector, row vector, and column vector
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| − | #** equal matrices
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| − | #** scalar multiplication
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| − | #** sum of matrices
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| − | #** zero matrix
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| − | #** scalar product
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| − | #** linear combination
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| − | #** matrix multiplication
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| − | #** transpose
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| − | #** trace
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| − | #** identity matrix
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| − | #* Perform the operations of matrix addition, scalar multiplication, transposition, trace, and matrix multiplication.
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| − | #* Represent matrices in terms of double subscript notation.
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| − | # Inverses; Rules of Matrix Arithmetic
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| − | #* Define the following:
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| − | #** commutative property
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| − | #** singular
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| − | #** nonsingular or invertible
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| − | #** multiplicative inverse
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| − | #* Recall, demonstrate, and apply algebraic properties for matrices.
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| − | #* Recall that matrix multiplication is not commutative in general. Determine conditions under which matrices do commute.
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| − | #* Recall and prove properties and identities involving the transpose operator.
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| − | #* Recall and prove properties and identities involving matrix inverses.
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| − | #* Recall and prove properties and identities involving matrix powers.
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| − | #* Recall, demonstrate, and apply that the cancelation laws for scalar multiplication do not hold for matrix multiplication.
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| − | #* Recall and apply the formula for the inverse of 2x2 matrices.
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| − | # Elementary Matrices
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| − | #* Define the following:
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| − | #** elementary matrix
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| − | #** row equivalent matrices
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| − | #* Identify elementary matrices and find their inverses or show that their inverse does not exist.
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| − | #* Relate elementary matrices to row operations.
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| − | #* Factor matrices using elementary matrices.
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| − | #* Find the inverse of a matrix, if possible, using elementary matrices.
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| − | #* Prove theorems about matrix products and matrix inverses.
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| − | #* Solve a linear equation using matrix inverses.
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| − | # Further Results on Systems of Equations and Invertibility
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| − | #* Solve matrix equations using matrix algebra.
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| − | #* Recall and prove properties and identities involving matrix inverses.
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| − | #* Recall equivalent conditions for invertibility.
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| − | # Further Results on Systems of Equations and Invertibility
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| − | #* Define the following:
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| − | #** diagonal matrix
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| − | #** upper and lower triangular matrices
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| − | #** symmetric matrix
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| − | #** skew-symmetric matrix
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| − | #* Determine powers of diagonal matrices.
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| − | #* Recall and prove properties and identities involving the transpose operator.
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| − | #* Prove basic facts involving symmetric and skew-symmetric matrices.
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| − | # Determinants
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| − | #* Define the following:
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| − | #** minor
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| − | #** cofactor
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| − | #** cofactor expansion
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| − | #** determinant
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| − | #** adjoint
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| − | #** Cramer's Rule
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| − | #* Apply cofactor expansion to evaluate determinants of nxn matrices.
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| − | #* Recall and apply the properties of determinants to evaluate determinants.
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| − | #* Evaluate the adjoint of a matrix.
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| − | #* Determine whether or not a matrix has an inverse based on its determinant.
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| − | #* Evaluate the inverse of a matrix using the adjoint method.
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| − | #* Use Cramer's rule to solve a linear system.
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| − | # Properties of Determinants
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| − | #* Recall the effects that row operations have on the determinants of matrices. Relate to the determinants of elementary matrices.
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| − | #* Recall, apply and verify the properties of determinants to evaluate determinants, including:
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| − | #** det(AB) = det(A) det(B)
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| − | #** det(kA) = k^n det(A)
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| − | #** det(A^-1)= 1/det(A)
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| − | #** det(A^T) = det(A)
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| − | #** det(A) = 0 if and only if A is singular
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| − | #* Evaluate the determinant of a matrix using row operations.
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| − | #* Apply determinants to determine invertibility of matrix products.
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| − | # Linear Transformations: Definitions and Examples
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| − | #* Define the following:
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| − | #** linear transformation
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| − | #** image
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| − | #** range
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| − | #* Describe geometrically the effects of a linear operator.
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| − | #* Determine whether or not a given transformation is linear.
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| − | #* Prove theorems and solve application problems involving linear transformations.
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| − | # Matrix Representations of Linear Transformations
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| − | #* Define the following:
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| − | #** standard matrix representation
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| − | #** eigenvalues and eigenvectors
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| − | #* Determine the matrix that represents a given linear transformation of vectors given an algebraic description.
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| − | #* Determine the matrix that represents a given linear transformation of vectors given a geometric description.
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| − | #* Prove theorems and solve application problems involving linear transformations.
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| − | # Vector Spaces: Definitions and Examples
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| − | #* Define the following:
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| − | #** vector space
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| − | #** vector space axioms
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| − | #** vector space R^n
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| − | #** vector space R^(mxn)
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| − | #** vector space of real-valued functions
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| − | #** additional properties of vector spaces
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| − | #* Prove or disprove that a given set of vectors together with an addition and a scalar multiplication is a vector space.
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| − | #* Prove and verify properties of a vector space.
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| − | # Subspaces
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| − | #* Define the following:
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| − | #** subspace
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| − | #** closure under addition
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| − | #** closure under scalar multiplication
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| − | #** zero subspace
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| − | #** linear combination
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| − | #** span (or subspace spanned by a set of vectors)
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| − | #** spanning set
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| − | #* Prove or disprove that a set of vectors forms a subspace.
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| − | #* Prove or disprove a set of vectors is a spanning set for R^n.
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| − | #* Prove or disprove a given vector is in the span of a set of vectors. Determine the span of a set of vectors.
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| − | #* Prove theorems about vector spaces and spans.
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| − | # Linear Independence
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| − | #* Define the following:
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| − | #** linearly independent
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| − | #** linearly dependent
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| − | #** Wronskian
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| − | #* Determine whether a set of vectors is linearly dependent or linearly independent.
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| − | #* Geometrically describe the span of a set of vectors. For sets that are linearly dependent, determine a dependence relation.
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| − | #* Prove theorems about linear independence.
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| − | # Basis and Dimension
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| − | #* Define the following:
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| − | #** basis
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| − | #** dimension
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| − | #** finite and infinite dimensional
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| − | #** standard basis
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| − | #* Prove or disprove a set of vectors forms a basis.
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| − | #* Find a basis for a vector space.
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| − | #* Determine the dimension of a vector space.
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| − | #* Geometrically interpret the ideas of span, linear dependance, basis, and dimension.
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| − | # Row Space, Column Space, and Null Space
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| − | #* Define the following:
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| − | #** row space
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| − | #** column space
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| − | #** null space
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| − | #** particular solution
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| − | #** general solution
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| − | #* Express a product Ax as a linear combination of column vectors.
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| − | #* Find a basis for a the column space, the row space, and the null space of a matrix.
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| − | #* Find the basis for a span of vectors.
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| − | # Rank and Nullity
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| − | #* Define the following:
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| − | #** rank
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| − | #** nullity
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| − | #** The Consistency Theorem
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| − | #** equivalent statements of invertibility
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| − | #* Find the rank and nullity of a matrix.
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| − | #* Recall and prove identities involving rank and nullity
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| − | #* Recall and apply the Consistency Theorm
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| − | #* Recall and apply the equivalent statements of invertibility.
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| − | # Eigenvalues and Eigenvectors
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| − | #* Define the following:
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| − | #** eigenvalue or characteristic value
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| − | #** eigenvector or characteristic vector
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| − | #** characteristic polynomial or characteristic polynomial
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| − | #** equivalent statements of invertibility
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| − | #* Find the eigenvalues and eigenvectors of an nxn matrix.
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| − | #* Prove theorems and solve application problems involving eigenvalues and eigenvectors.
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| − | # Diagonalization
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| − | #* Define the following:
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| − | #** diagonalizable
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| − | #** algebraic multiplicity
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| − | #** geometric multiplicity
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| − | #* Determine whether or not a matrix is diagonalizable.
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| − | #* Find the diagonalization of a matrix, if possible.
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| − | #* Find powers of a matrix using the diagonalization of a matrix.
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| − | #* Prove theorems and solve application problems involving the diagonalization of matrices.
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| − | # Limit, Continuity, Vector Derivative; The Rules of Differentiation
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| − | #* Define the following:
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| − | #** scalar functions
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| − | #** vector functions
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| − | #** components of a vector function
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| − | #** plane curve or space curve
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| − | #** parametrization of a curve
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| − | #** limit of a vector function
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| − | #** a vector function continuous at a point
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| − | #** derivative of a vector function
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| − | #** a differentiable vector function
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| − | #** integral of a vector function
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| − | #* Graph a parametric curve.
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| − | #* Identify a curve given its parametrization.
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| − | #* Determine combinations of vector functions such as sums, vector products and scalar products.
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| − | #* Evaluate limits, derivatives, and integrals of vector functions.
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| − | #* Recall, derive and apply rules to combinations of vector functions for the following:
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| − | #** limits
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| − | #** differentiation
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| − | #** integration
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| − | #* Determine continuity of a vector-valued function.
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| − | #* Prove theorems involving limits and derivatives of vector-valued functions.
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| − | #* Solve application problems involving vector-valued functions.
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| − | # Curves; Vector Calculus in Mechanics
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| − | #* Define the following:
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| − | #** directed path
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| − | #** differentiable parameterized curve
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| − | #** tangent vector
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| − | #** tangent line
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| − | #** unit tangent vector
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| − | #** principal normal vector
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| − | #** normal line
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| − | #** osculation plane
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| − | #** force vector
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| − | #** momentum vector
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| − | #** angular momentum vector
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| − | #** torque
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| − | #* Find the tangent vector and tangent line to a curve at a given point.
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| − | #* Find the principle normal and normal line to a curve at a given point.
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| − | #* Determine the osculating plane for a space curve at a given point.
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| − | #* Reverse the direction of a curve.
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| − | #* Solve application problems involving curves.
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| − | #* Solve application problems involving force, momentum, angular momentum, and torque.
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| − | # Arc Length
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| − | #* Define the following:
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| − | #** arc length
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| − | #** arc length parametrization
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| − | #* Evaluate the arc length of a curve.
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| − | #* Determine whether a curve is arc length parameterized.
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| − | #* Find the arc length parametrization of a curve.
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| − | # Curvilinear Motion; Curvature
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| − | #* Define the following:
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| − | #** velocity vector function
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| − | #** speed
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| − | #** acceleration vector function
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| − | #** uniform circular motion
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| − | #** curvature
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| − | #** tangential component of acceleration
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| − | #** normal component of acceleration
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| − | #* Given the position vector function of a moving object, calculate the velocity vector function, speed, and acceleration vector function, and vice versa.
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| − | #* Calculate the curvature of a space curve.
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| − | #* Recall the formulas for the curvature of a parameterized planar curve or a planar curve that is the graph of a function. Apply these formulas to calculate the curvature of a planar curve.
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| − | #* Determine the tangential and normal components of acceleration for a given parameterized curve.
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| − | #* Solve application problems involving curvilinear motion and curvature.
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| − | # Functions of Several Variables; A Brief Catalogue of the Quadric Surfaces; Projections
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| − | #* Define the following:
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| − | #** real-valued function of several variables
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| − | #** domain
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| − | #** range
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| − | #** bounded functions
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| − | #** quadric surface
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| − | #** intercepts
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| − | #** traces
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| − | #** sections
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| − | #** center
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| − | #** symmetry
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| − | #** boundedness
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| − | #** cylinder
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| − | #** ellipsiod
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| − | #** elliptic cone
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| − | #** elliptic paraboloid
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| − | #** hyperboloid of one sheet
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| − | #** hyperboloid of two sheets
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| − | #** hyperbolic paraboloid
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| − | #** parabolic cylinder
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| − | #** elliptic cylinder
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| − | #** projection of a curve onto a coordinate plane
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| − | #* Describe the domain and range of a function of several variables.
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| − | #* Write a function of several variables given a description.
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| − | #* Identify standard quadratic surfaces given their functions or graphs.
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| − | #* Sketch the graph of a quadratic surface by sketching intercepts, traces, sections, centers, symmetry, boundedness.
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| − | #* Find the projection of a curve, that is the intersection of two surfaces, to a coordinate plane.
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| − | # Graphs; Level Curves and Level Surfaces
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| − | #* Define the following:
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| − | #** level curve
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| − | #** level surface
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| − | #* Describe the level sets of a function of several variables.
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| − | #* Graphically represent a function of two variables by level curves or a function of three variables by level surfaces.
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| − | #* Identify the characteristics of a function from its graph or from a graph of its level curves (or level surfaces).
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| − | #* Solve application problems involving level sets. functions.
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| − | # Partial Derivatives
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| − | #* Define the following:
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| − | #** partial derivative of a function of several variables
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| − | #** second partial derivative
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| − | #** mixed partial derivative
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| − | #* Interpret the definition of a partial derivative of a function of two variables graphically.
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| − | #* Evaluate the partial derivatives of a function of several variables.
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| − | #* Evaluate the higher order partial derivatives of a function of several variables.
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| − | #* Verify equations involving partial derivatives.
| |
| − | #* Apply partial derivatives to solve application problems.
| |
| − | # Open and Closed Sets; Limits and Continuity; Equity of Mixed Partials
| |
| − | #* Define the following:
| |
| − | #** neighborhood of a point
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| − | #** deleted neighborhood of a point
| |
| − | #** interior of a set
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| − | #** boundary of a set
| |
| − | #** open set
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| − | #** closed set
| |
| − | #** limit of a function of several variables at a point
| |
| − | #** continuity of a function of several variables at a point
| |
| − | #* Determine the boundary and interior of a set.
| |
| − | #* Determine whether a set is open, closed, neither, or both.
| |
| − | #* Evaluate the limit of a function of several variables or show that it does not exists.
| |
| − | #* Determine whether or not a function is continuous at a given point.
| |
| − | #* Recall and apply the conditions under which mixed partial derivatives are equal.
| |
| − | # Differentiability and Gradient
| |
| − | #* Define the following:
| |
| − | #** differentiable multivariable function
| |
| − | #** gradient of a multivariable function
| |
| − | #* Evaluate the gradient of a function.
| |
| − | #* Find a function with a given gradient.
| |
| − | # Gradient and Directional Derivative
| |
| − | #* Define the following:
| |
| − | #** directional derivative
| |
| − | #** isothermals
| |
| − | #* Recall and prove identities involving gradients.
| |
| − | #* Give a graphical interpretation of the gradient.
| |
| − | #* Evaluate the directional derivative of a function.
| |
| − | #* Give a graphical interpretation of directional derivative.
| |
| − | #* Recall, prove, and apply the theorem that states that a differential function f increases most rapidly in the direction of the gradient (the rate of change is then ||f(x)||) and it decreases most rapidly in the opposite direction (the rate of change is then -||f(x)||).
| |
| − | #* Find the path of a heat seeking or a heat repelling particle.
| |
| − | #* Solve application problems involving gradient and directional derivatives.
| |
| − | # The Mean-Value Theorem; The Chain Rule
| |
| − | #* Define the following:
| |
| − | #** the Mean Value Theorem for functions of several variables
| |
| − | #** normal line
| |
| − | #** chain rules for functions of several variables
| |
| − | #** implicit differentiation
| |
| − | #* Recall and apply the Mean Value Theorem for functions of several variables and its corollaries.
| |
| − | #* Apply an appropriate chain rule to evaluate a rate of change.
| |
| − | #* Apply implicit differentiation to evaluate rates of change.
| |
| − | #* Solve application problems involving chain rules and implicit differentiation.
| |
| − | # The Gradient as a Normal; Tangent Lines and Tangent Planes
| |
| − | #* Define the following:
| |
| − | #** normal vector
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| − | #** tangent vector
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| − | #** tangent line
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| − | #** tangent plane
| |
| − | #** normal line
| |
| − | #* Use gradients to find the normal vector and normal line to a smooth planar curve at a given point.
| |
| − | #* Use gradients to find the tangent vector and tangent line to a smooth planar curve at a given point.
| |
| − | #* Use gradients to find the normal vector to a smooth surface at a given point.
| |
| − | #* Use gradients to find the tangent plane to a smooth surface at a given point.
| |
| − | #* Use gradients to find the normal line to a smooth surface at a given point.
| |
| − | #* Solve application problems involving normals and tangents to curves and surfaces.
| |
| − | # Local Extreme Values
| |
| − | #* Define the following:
| |
| − | #** local minimum and local maximum
| |
| − | #** critical points
| |
| − | #** stationary points
| |
| − | #** saddle points
| |
| − | #** discriminant
| |
| − | #** Second Derivative Test
| |
| − | #* Find the critical points of a function of two variables.
| |
| − | #* Apply the Second-Partials Test to determine whether each critical point is a local minimum, a local maximum, or a saddle point.
| |
| − | #* Solve word problems involving local extreme values.
| |
| − | # Absolute Extreme Values
| |
| − | #* Define the following:
| |
| − | #** absolute minimum and absolute maximum
| |
| − | #** bounded subset of a plane or three-space
| |
| − | #** the Extreme Value Theorem
| |
| − | #* Determine absolute extreme values of a function defined on a closed and bounded set.
| |
| − | #* Apply the Extreme Value Theorem to justify the method for finding extreme values of functions defined on certain sets.
| |
| − | #* Solve word problems involving absolute extreme values.
| |
| − | # Maxima and Minima with Side Conditions
| |
| − | #* Define the following:
| |
| − | #** side conditions or constraints
| |
| − | #** method of Lagrange
| |
| − | #** Lagrange multipliers
| |
| − | #** cross-product equation of the Lagrange condition
| |
| − | #* Graphically interpret the method of Lagrange.
| |
| − | #* Determine the extreme values of a function subject to a side conditions by applying the method of Lagrange.
| |
| − | #* Apply the cross-product equation of the Lagrange condition to solve extreme value problems subject to side conditions.
| |
| − | #* Apply the method of Lagrange to solve word problems.
| |
| − | # Differentials; Reconstructing a Function from its Gradient
| |
| − | #* Define the following:
| |
| − | #** differential
| |
| − | #** general solution
| |
| − | #** particular solution
| |
| − | #** connected open set
| |
| − | #** open region
| |
| − | #** simple closed curve
| |
| − | #** simply connected open region
| |
| − | #** partial derivative gradient test
| |
| − | #* Determine the differential for a given function of several variables.
| |
| − | #* Determine whether or not a vector function is a gradient.
| |
| − | #* Given a vector function that is a gradient, find the functions with that gradient.
| |
| − | # Multiple-Sigma Notation; The Double Integral over a Rectangle R; The Evaluation of Double Integrals by Repeated Integrals
| |
| − | #* Define the following:
| |
| − | #** double sigma notation
| |
| − | #** triple sigma notation
| |
| − | #** upper sum
| |
| − | #** lower sum
| |
| − | #** double integral
| |
| − | #** integral formula for the volume of a solid bounded between a region Omega in the xy-plane and the graph of a non-negative function z = f(x,y) defined on Omega.
| |
| − | #** integral formula for the area of region in a plane
| |
| − | #** integral formula for the average of a function defined on a region Omega.
| |
| − | #** projection of a region onto a coordinate axis
| |
| − | #** Type I and Type II regions
| |
| − | #** reduction formulas for double integrals
| |
| − | #** the geometric interpretation of the reduction formulas for double integrals
| |
| − | #* Evaluate double and triple sums given their sigma notation.
| |
| − | #* Recall and apply summation identities.
| |
| − | #* Approximate the integral of a function by a lower sum and an upper sum.
| |
| − | #* Evaluate the integral of a function using the definition.
| |
| − | #* Evaluate double integrals over a rectangle using the reduction formulas.
| |
| − | #* Sketch planar regions and determine if they are Type I, Type II, or both.
| |
| − | #* Evaluate double integrals over Type I and Type II regions.
| |
| − | #* Change the order of integration of an integral.
| |
| − | #* Apply double integrals to calculate volumes, areas, and averages.
| |
| − | # The Double Integral as the Limit of Riemann Sums; Polar Coordinates
| |
| − | #* Define the following:
| |
| − | #** diameter of a set
| |
| − | #** Riemann sum
| |
| − | #** double integral as a limit of Riemann sums
| |
| − | #** polar coordinates (r; theta)
| |
| − | #** transformation formulas between Cartesian and polar coordinates
| |
| − | #** double integral conversion formula between Cartesian and polar coordinates
| |
| − | #* Represent a region in both Cartesian and polar coordinates.
| |
| − | #* Evaluate double integrals in terms of polar coordinates.
| |
| − | #* Evaluate areas and volumes using polar coordinates.
| |
| − | #* Convert a double integral in Cartesian coordinates to a double integral in polar coordinates and then evaluate.
| |
| − | # Further Applications of the Double Integral
| |
| − | #* Define the following:
| |
| − | #** integral formula for the mass of a plate
| |
| − | #** integral formulas for the center of mass of a plate
| |
| − | #** integral formulas for the centroid of a plate
| |
| − | #** integral formulas for the moment of an inertia of a plate
| |
| − | #** radius of gyration
| |
| − | #** the Parallel Axis Theorem
| |
| − | #* Evaluate the mass and center or mass of a plate
| |
| − | #* Evaluate the centroid of a plate.
| |
| − | #* Evaluate the moments of inertia of a plate.
| |
| − | #* Calculate the radius of gyration of a plate.
| |
| − | #* Recall and apply the parallel axis theorem.
| |
| − | # Triple Integrals; Reduction to Repeated Integrals
| |
| − | #* Define the following:
| |
| − | #** triple integral
| |
| − | #** integral formula for the volume of a solid
| |
| − | #** integral formula for the mass of a solid
| |
| − | #** integral formulas for the center of mass of a solid
| |
| − | #* Evaluate physical quantities using triple integrals such as volume, mass, center of mass, and moments of intertia.
| |
| − | #* Recall and apply the properties of triple integrals, including: linearity, order, additivity, and the mean-value condition.
| |
| − | #* Sketch the domain of integration of an iterated integral.
| |
| − | #* Change the order of integration of a triple integral.
| |
| − | # Cylindrical Coordinates
| |
| − | #* Define the following:
| |
| − | #** cylindrical coordinates of a point
| |
| − | #** coordinate transformations between Cartesian and cylindrical coordinates
| |
| − | #** cylindrical element of volume
| |
| − | #* Convert between Cartesian and cylindrical coordinates.
| |
| − | #* Describe regions in cylindrical coordinates.
| |
| − | #* Evaluate triple integrals using cylindrical coordinates.
| |
| − | # Spherical Coordinates
| |
| − | #* Define the following:
| |
| − | #** spherical coordinates of a point
| |
| − | #** coordinate transformations between Cartesian and spherical coordinates
| |
| − | #** spherical element of volume
| |
| − | #* Convert between Cartesian and spherical coordinates.
| |
| − | #* Describe regions in spherical coordinates.
| |
| − | #* Evaluate triple integrals using spherical coordinates.
| |
| − | # Jacobians; Changing Variables in Multiple Integration
| |
| − | #* Define the following:
| |
| − | #** Jacobian
| |
| − | #** change of variable formula for double integration
| |
| − | #** change of variable formula for triple integration
| |
| − | #* Find the Jacobian of a coordinate transformation.
| |
| − | #* Use a coordinate transformation to evaluate double and triple integrals.
| |
| − | # Line Integrals
| |
| − | #* Define the following:
| |
| − | #** work along a curved path
| |
| − | #** smooth parametric curve
| |
| − | #** directed or oriented curve
| |
| − | #** path dependence
| |
| − | #** closed curve
| |
| − | #* Evaluate the work done by a varying force over a curved path.
| |
| − | #* Evaluate line integrals in general including line integrals with respect to arc length.
| |
| − | #* Evaluate the physical characteristics of a wire such as centroid, mass, and center of mass using line integrals.
| |
| − | #* Determine whether or not a vector field is a gradient.
| |
| − | #* Determine whether or not a differential form is exact.
| |
| − | # The Fundamental Theorem for Line Integrals; Work-Energy Formula; Conservation of Mechanical Energy
| |
| − | #* Define the following:
| |
| − | #** path-independent line integrals
| |
| − | #** closed vector field
| |
| − | #** simply connected
| |
| − | #* Recall, apply, and verify the Fundamental Theorem for Line Integrals (Theorem 2 in Section 15.3).
| |
| − | #* Determine whether or not a force field is closed on a given region, and if so, find its potential function.
| |
| − | #* Solve application problems involving work done by a conservative vector field
| |
| − | # Vector Fields
| |
| − | #* Define the following:
| |
| − | #** vector field
| |
| − | #** open
| |
| − | #** path connected
| |
| − | #** region
| |
| − | #** integral curve (field lines, flow lines, or streamlines)
| |
| − | #** gradient vector field (or conservative vector field)
| |
| − | #** potential function
| |
| − | #** continuously differentiable vector field
| |
| − | #* Sketch a vector field.
| |
| − | #* Write the formula for a vector field from a description.
| |
| − | #* Write the gradient vector field associated with a given scalar-valued function.
| |
| − | #* Recover a function from its gradient or show it is not possible.
| |
| − | #* Find the integral curves of a vector field.
| |
| − | # Green's Theorem
| |
| − | #* Define the following:
| |
| − | #** Jordan curve
| |
| − | #** Jordan region
| |
| − | #** Green's Theorem
| |
| − | #* Recall and verify Green's Theorem.
| |
| − | #* Apply Green's Theorem to evaluate line integrals.
| |
| − | #* Apply Green's Theorem to find the area of a region.
| |
| − | #* Derive identities involving Green's Theorem
| |
| − | # Parameterized Surfaces; Surface Area
| |
| − | #* Define the following:
| |
| − | #** parameterized surface
| |
| − | #** fundamental vector product
| |
| − | #** element of surface area for a parameterized surface
| |
| − | #** surface integral
| |
| − | #** integral formula for the surface area of a parameterized surface
| |
| − | #** integral formula for the surface area of a surface z = f(x; y)
| |
| − | #** upward unit normal
| |
| − | #* parameterize a surface.
| |
| − | #* evaluate the fundamental vector product for a parameterized surface.
| |
| − | #* Calculate the surface area of a parameterized surface.
| |
| − | #* Calculate the surface area of a surface z = f(x; y).
| |
| − | # Surface Integrals
| |
| − | #* Define the following:
| |
| − | #** surface integral
| |
| − | #** integral formulas for the surface area and centroid of a parameterized surface
| |
| − | #** integral formulas for the mass and center of mass of a parameterized surface
| |
| − | #** integral formulas for the moments of inertia of a parameterized surface
| |
| − | #** integral formula for flux through a surface
| |
| − | #* Calculate the surface area and centroid of a parameterized surface.
| |
| − | #* Calculate the mass and center of mass of a parameterized surface.
| |
| − | #* Calculate the moments of inertia of a parameterized surface.
| |
| − | #* Evaluate the flux of a vector field through a surface.
| |
| − | #* Solve application problems involving surface integrals.
| |
| − | # The Vector Differential Operator Del
| |
| − | #* Define the following:
| |
| − | #** the vector differential operator Del
| |
| − | #** divergence
| |
| − | #** curl
| |
| − | #** Laplacian
| |
| − | #* Evaluate the divergence of a vector field.
| |
| − | #* Evaluate the curl of a vector field
| |
| − | #* Evaluate the Laplacian of a function.
| |
| − | #* Recall, derive and apply formulas involving divergence, gradient and Laplacian.
| |
| − | #* Interpret that divergence and curl of a vector fields physically.
| |
| − | # The Divergence Theorem
| |
| − | #* Define the following:
| |
| − | #** outward unit normal
| |
| − | #** the divergence theorem
| |
| − | #** sink and source
| |
| − | #** solenoidal
| |
| − | #* Recall and verify the Divergence Theorem.
| |
| − | #* Apply the Divergence Theorem to evaluate the flux through a surface.
| |
| − | #* Solve application problems using the Divergence Theorem.
| |
| − | # Stokes' Theorem
| |
| − | #* Define the following:
| |
| − | #** oriented surface
| |
| − | #** outward, upward, and downward unit normal
| |
| − | #** the positive sense around the boundary of a surface
| |
| − | #** circulation
| |
| − | #** component of curl in the normal direction
| |
| − | #** irrotational
| |
| − | #** Stokes' theorem
| |
| − | #* Recall and verify Stoke's theorem.
| |
| − | #* Use Stokes' Theorem to calculate the flux of a curl vector field through a surface by a line integral.
| |
| − | #* Apply Stokes' theorem to calculate the work (or circulation) of a vector field around a simple closed curve.
| |
| − |
| |
| − |
| |
| − |
| |
| − |
| |
| − | </div>
| |
| − |
| |
| − | === Textbooks ===
| |
| − | Possible textbooks for this course include (but are not limited to):
| |
| − |
| |
| − | *
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| − |
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| − |
| |
| − | === Additional topics ===
| |
| − |
| |
| − | === Courses for which this course is prerequisite ===
| |
| − | [[Math 303 Mathematics for Engineering 2|Math 303]]
| |
| − |
| |
| − | [[Category:Courses|302]]
| |
