## topics

topics

### Geometry of space; vectors

Geometry of space; vectors

- Use Cartesian, cylindrical, and spherical coordinates, and be able to convert between the different coordinate systems.
- Resolve geometric vectors into components parallel to coordinate axes.
- Perform the operations of vector addition and scalar multiplication, and interpret them geometrically.
- Use the dot product to calculate magnitude of a vector, angle between vectors, and projection of one vector on another.
- Use the cross product; interpret the cross product geometrically and as area of a parallelogram.
- Use and interpret geometrically the standard equation for a plane, and the parametric and vector forms for a line.
- Use parametric equations for curves; find tangent lines to parametric curves; find length of parametric curves.
- Compute velocity, unit tangent and acceleration vectors along a parametric curve.
- Resolve acceleration into tangential and normal components; compute curvature.
- Recognize cylinders and quadric surfaces from their Cartesian equations.

### Differentiation for functions of two or more variables

Differentiation for functions of two or more variables

- Represent a function of two variables as the graph of a surface; sketch level curves; level surfaces.
- Calculate partial derivatives. Find tangent planes to functions of two variables.
- Use the Chain Rule; implicit differentiation of multi-variable functions.
- Find the gradient and interpret the gradient geometrically. Use the gradient to find tangent planes.
- Compute directional derivatives.
- Use linear approximations to estimate the function near a point of tangency.
- Find second-order Taylor polynomials for multi-variable functions.
- Find and classify critical points of functions, using the second partials test.
- Find maximum and minimum values for a function defined on a closed and bounded set.
- Use the method of Lagrange multipliers in optimization.

### Integration for multi-variable functions and vector calculus

Integration for multi-variable functions and vector calculus

- Use iterated integrals to evaluate double integrals over regions in the plane.
- Set up and evaluate double integrals in polar coordinates.
- Set up and evaluate triple integrals over regions in space using Cartesian coordinates.
- Set up and evaluate integrals to compute area, surface area, volume, mass, moments, center of mass, and inertia.
- Set up and evaluate integrals in cylindrical and spherical coordinates.
- Change the order of integration in multiple integrals.
- Carry out change of variables (Jacobian) in multiple integrals.
- Set up and evaluate line integrals of scalar functions or vector fields along curves. Compute work along a line.
- Determine the curl and divergence of a vector field.
- Recognize conservative vector fields, and apply the fundamental theorem for line integrals of conservative vector fields.
- State and apply Green's Theorem.
- Set up and evaluate surface integrals; compute the flux of a vector field through a boundary.
- (*)State and apply Stokes' Theorem.
- (*)State and apply the Divergence Theorem.

(*) = Topic covered during dead week; not on Exam 3