# Exam reviews

(Note coverage of topics has moved a bit over the years)

## Geometry of space; vectors

### Major topics include:

• Cartesian, cylindrical, and spherical coordinate systems
• Using vectors: addition, scaling, component form
• Dot product; angles between vectors; projection
• Cross product; finding perpendicular vectors; areas/volumes
• Planes and lines
• Parametric equations for curves; tangent line
• Velocity and acceleration; decomposition of acceleration
• Length of parametric curves

## Multivariable differentiation

### Major topics include:

• Functions of two (or more) variables as graphs; level curves and level surfaces
• Compute partial derivatives
• Find tangent plane to functions of two variables
• Chain rule; implicit differentiation
• Gradient vector and its geometrical properties; tangent planes to implicit surfaces
• Directional derivatives
• Linear approximation
• Second-order Taylor polynomials for multi-variable functions
• Finding critical points; classifying critical points using second partials test
• Absolute max and min of a continuous function on closed and bounded set
• Lagrange multipliers method for optimization with constraint

## Multivariable integration

### Major topics include:

• Iterated integration; changing order of 2D integrals
• Integration in polar coordinates
• Triple integrals over 3D regions; changing order of 3D integrals
• Using integration to find area, surface area, volume, mass, moments, center of mass, and inertia
• Integrate in cylindrical and spherical coordinate systems
• Change of variables (Jacobian)
• Vector fields; curl and divergence of vector fields
• Line integrals; work along a line; conservative line integrals
• Green's Theorem
• Surface integrals; flux through a surface
• Stokes' Theorem
• Divergence (Gauss') Theorem