MATH 2220 - Multivariable Calculus and Vector Analysis

1. Sketch the curve determined by a vector-valued function and indicate the orientation.
2. Find the arc length of a parametrized curve.
3. Find the domain, limit, derivative, and integrals of vector-valued functions.
4. Find parametric equations for the tangent line to a curve at a point.
5. Find an equation of the normal plane to a given curve at a point.
6. Find the unit tangent and normal vectors for the curve determined by a vector-valued function.
7. Find the curvature of a curve at a point and the radius of curvature.
8. Find the tangential and normal components of acceleration, and the curvature of the curve determined by a vector-valued function.
9. Find the domain of functions with several variables.
10. Sketch and describe level curves of a function.
11. Find an equation of the level curve or level surface of a function that contains a given point.
12. Find limits of functions of several variables.
13. Describe the set of all points at which a function of several variables is continuous.
14. Find partial derivatives, including the use of the chain rule.
15. Find differentials to find the approximate change in a function.
16. Find the directional derivatives of a function at a point in an indicated direction.
17. Find the gradient of a function at a point.
18. Find the equations for the tangent plane and the normal line to the graph of an equation at a point.
19. Find the extrema of functions of several variables on both the interior and boundary of its domain.
20. Use La’Grange multipliers to find the extrema of a function subject to stated constraints.
21. Solve multiple integrations problems; find area and volume using multiple integration.
22. Solve double integrals in polar coordinates.
23. Find the surface area of the portion of the graph that lies over a region in the xy-plane.
24. Find the mass and the center of mass of the lamina that has the shape of the region bounded by the graphs of given equations and has the given area mass density.
25. Work with cylindrical and spherical coordinates. Change from xyz-coordinates to either cylindrical or spherical. Evaluate triple integrals by changing to cylindrical or spherical coordinates.
26. If T is a transformation from the xy-plane to the uv-plane determined by given formulas, describe the u-curves and the v-curves, find formulas in terms of u and v that specify T^-1, and the curve in the uv-plane that corresponds to a given part of a curve in the xy-plane. Find Jacobians. Evaluate integrals by making indicated change of variables.
27. Evaluate line integrals along a curve by definition and with the Fundamental Theorem of Line Integrals for conservative vector fields.
28. Show that a line integral is independent or not independent of path.
29. Use Green’s Theorem to evaluate line integrals.
30. Evaluate surface integrals.
31. Use the Divergence Theorem to evaluate surface integrals, the flux of a vector field through a surface.
32. Apply Stokes’ Theorem.

1. Vector-Valued Functions
2. Partial Differentiation
3. Multiple Integrals
4. Vector Calculus

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