Upon successful completion of this course, students will:

Perform calculus operations on vector valued functions, including derivatives, integrals, curvature, displacement, velocity, acceleration, and torsion by
1. finding the domain of a vector valued function;
2. sketching the graph of a vector valued function;
3. evaluating the limit of a vector valued function;
4. finding the derivative of a vector valued function;
5. finding the tangent vector to the graph of a vector valued function;
6. finding the indefinite integral of a vector valued function;
7. finding the definite integral of a vector valued function;
8. finding the arc length of a vector valued function over a given interval;
9. writing a vector valued function in terms of an arc length parameter;
10. finding the curvature of a vector valued function.
Perform calculus operations on functions of several variables, including partial derivatives, directional derivatives, and multiple integrals by
1. sketching the domain;
2. evaluating at an element in the domain;
3. sketching the level curves or level surfaces;
4. evaluating the limit;
5. determining where a function is continuous;
6. determining partial derivatives;
7. finding the differential;
8. determining the derivative of composite functions using the chain rule;
9. finding the gradient;
10. using the gradient to find a directional derivative;
11. evaluating a double integral over a type 1 or type 2 region;
12. using a double integral to find the area of a region in the xy-plane;
13. evaluating double integrals using polar coordinates;
14. evaluating a triple integral in rectangular coordinates;
15. evaluating a triple integral in cylindrical coordinates;
16. evaluating a triple integral in spherical coordinates;
17. evaluating a double integral by performing a change of variable.
Find extrema and tangent planes by
1. finding the tangent plane to a surface at a point;
2. finding any relative minima, relative maxima, or saddle points;
3. solving an optimization problem using Lagrange multipliers.
Solve problems using the Fundamental Theorem of Line Integrals, Green's Theorem, the Divergence Theorem, and Stokes' Theorem by
1. defining a vector field;
2. producing graphs of vector fields;
3. evaluating line integrals;
4. evaluating integrals using the Fundamental Theorem of Line Integrals;
5. determining if a line integral is independent of path;
6. using Green’s Theorem to evaluate integrals;
7. evaluating surface integrals;
8. evaluating integrals using the Divergence Theorem;
9. evaluating integrals using Stokes’ Theorem.
Apply the computational and conceptual principles of calculus to the solutions of real world problems by
1. determining displacement, velocity, and acceleration vectors to analyze motion along curves in 2 and 3 dimensional space;
2. solving application problems involving the gradient;
3. finding the flux, the divergence and the curl of a vector field;
4. solving work problems in a vector field using line integrals.
Demonstrate an understanding of 3 dimensional analytic geometry by
1. producing a plot of points;
2. finding the distance between 2 points;
3. finding the midpoint of a line segment;
4. writing the equation of a sphere;
5. writing the parametric equations of a line;
6. writing the equation of a plane;
7. producing a graph of a cylindrical surface;
8. producing a graph of a quadric surface.
Demonstrate an understanding of the properties of vectors by
1. producing the graph of a vector in standard position;
2. calculating the magnitude of a vector;
3. converting a vector to a unit vector;
4. finding the sum of two vectors;
5. finding a scalar multiple of a vector;
6. finding the dot product of two vectors;
7. finding the cross product of two vectors;
8. finding the angle between two vectors;
9. finding the projection of one vector onto another;
10. determining if two vectors are orthogonal;
11. determining if two vectors are parallel;
12. finding the triple scalar product of three vectors.

MATH 2415

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MATH 2415 - Calculus III