MATH& 254 - Calculus IV

1. Calculate limits and determine continuity of multivariable functions.
2. Evaluate partial derivatives and apply the chain rule.
3. Calculate the gradient and use it to find the equations of tangent lines and planes.
4. Determine extrema of a multivariable function by applying second derivative tests.
5. Solve optimization applications using Lagrange Multipliers.
6. Calculate areas and volume of solids using iterated integrals.
7. Change the variables of integration using the Jacobian, including cylindrical and spherical coordinates.
8. Calculate line integrals over vector fields using the Fundamental Theorem for Line Integrals.
9. Use vector and scalar fields appropriately, particularly when computing a gradient, curl or divergence using the differential vector operator “del”.
10. Calculate surface integrals over oriented surfaces and over vector fields.
11. Perform calculations using Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem.

1. Functions of Several Variables. Limits and continuity. Partial Derivatives. The Chain Rule. Directional derivatives and the gradient vector. Tangent planes and linear approximations.
2. Multiple Integrals. Iterated integrals including double and triple integrals. Change of order of integration. Triple integrals in cylindrical and spherical coordinates. Change of variables and the Jacobian. Applications including area, surface area, volumes. Calculate density and mass. Calculate moments and centers of mass. Calculate moments of inertia.
3. Vector Calculus. Vector fields. Line integrals and the Fundamental Theorem for Line Integrals. Curl and Divergence. Surface Integrals. Surface integrals of vector fields. Green’s Theorem. Stoke’s Theorem. The Divergence Theorem.