1. Determine the rectangular, cylindrical, or spherical coordinates of a point given the coordinates of the point in any one of those systems.
2. For two points in space, calculate the distance between those points and their midpoint, and determine a parameterization or vector equation of the line that contains those points.
3. Given two vectors in space, calculate: any linear combination of those vectors, the measure of the angle formed between those vectors, the area of the triangle framed by those vectors, the vector resulting from projecting one vector onto the other, and a vector that is orthogonal to both of of those vectors.
4. Determine an equation for the plane that contains three non-colinear points given their coordinates
5. Determine the line along which two planes intersect.
6. Determine the point at which a line intersects a plane.
7. Compute the distance to a plane from a point.
8. Compute the distance to a line from a point.
9. Compute the distance between two skew lines.
10. Determine the line tangent to a curve at a given point.
11. Calculate the unit tangent, unit normal, and unit binormal vectors to a parameterization of a curve at a point.
12. Calculate the curvature and the torsion of a parameterization of a curve at a point.
13. Sketch a contour plot and graph for a multivariable function.
14. Prove that the limit of a multivariable function \( \lim_{(x,y)\to(a,b)} f(x,y) \) either exists or doesn’t exist, and if it exists compute its value.
15. Compute the partial derivatives of a multivariable function.
16. Given a surface defined either parametrically or as the graph \(z = f(x,y)\) of a multivariable function, determine an equation of the plane tangent to the surface at a point .
17. Given an equation involving the multiple variables, compute the partial derivative of any of them with respect to one of the others, both in the case that those symbols represent independent variables and in the case that they represent functions of other independent variables. If possible, be able to solve this implicit partial derivative to express the partial derivative explicitly.
18. Compute the gradient vector of a multivariable function, and the directional derivative of that function in a given direction.
19. Determine the extreme values of a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) on an open subset of the interior of its domain and on a closed subset of the boundary of its domain.
20. Given a region \(R\) in \(\mathbf{R}^2\) with a boundary that can be described analytically either in rectangular or polar coordinates, and a function \(f\colon R \to \mathbf{R},\) compute the value of \( \iint_R f \,\mathrm{d}A\,. \)
21. Given an expanse \(E\) in \(\mathbf{R}^3\) with a boundary that can be described analytically either in rectangular, cylindrical, or spherical coordinates, and a function \(f\colon E \to \mathbf{R},\) compute the value of \( \iiint_E f \,\mathrm{d}V\,. \)
22. Determine whether or not a vector field over either \(\mathbf{R}^2\) or \(\mathbf{R}^3,\) is conservative, and if so determine its potential function.
23. Compute the divergence and curl of a vector field over either \(\mathbf{R}^2\) or \(\mathbf{R}^3.\)
24. Evaluate a line integral over a piecewise-smooth curve in a vector field. If the curve is closed, compute its value either directly or as an integral over its interior.
25. Evaluate a surface integral over a piecewise-smooth surface in a vector field. If the surface is closed, compute its value either directly or as an integral over its interior.