Below is a list of exercises from Calculus 9E by Stewart. These are only “exercises” though, and not “problems”; while they provide good practice at the computations that are the foundation of working with calculus in higher dimensions, they are only a foundation, a means to being able to solve problems that earnestly imbue one with understanding; remember that understanding is the goal.
Note that the numerical answers to odd-numbered exercises can be found in the back of the textbook.
- Become familiar with three-dimensional space. Specifically know where points are located (1–10), be able to calculate the distance between two points (11–14), understand the equation and geometry of spheres (15–26), and be able to translate between and equation/inequality and the region it’s describing (27–46). Also convince yourself that the three-dimensional distance formula really does follow from applying the two-dimensional distance formula twice.
- Get good at visualizing vectors (1–14), fluent in the notation of vectors, and performing basic operations with them (15–28).
- Calculate dot products (2–10), understand the dot product’s relationship to the angle subtended by two vectors (15–30), and calculate the projection of one vector onto another (39–46).
- Get comfortable computing cross products (1–12), get comfortable with their properties (23–26), and understand their underlying geometry (16–22, 27–38).
- Get comfortable with the geometry of lines and planes in space (1, 41–44), and get comfortable with the algebra underlying the parametric-equations that represent lines and planes (2–40, 45–60). If some of these exercises seem tough, come back to them after we’ve discussed vectors.
- Be able to sketch the surface corresponding to a quartic equation, and reverse-engineer the equation from a given surface (1–30), and remember how to “complete the square” for the sake of algebraically manipulate quartic equations (33–40).
- Be able to sketch space curves and their projections (7–20, 25–34).
- Recall how to compute derivatives and antiderivatives. Take the derivatives of vector-valued functions and understand what those derivatives represent geometrically (1–28) .
- Compute arclength (1–11, 17), and compute curvature (19–35).
- Compute velocity and acceleration (3–16).
- Get comfortable with the mechanics of multivariable functions (1–16), be able sketch the graphs and level curves of \(\mathbf{R}^2 \to \mathbf{R}\) functions (23–33, 36, 41–52, 61–66), and be able to sketch the level surfaces of \(\mathbf{R}^3 \to \mathbf{R}\) functions (67–70).
- Evaluate limits, determine that a limit doesn’t exist (5–30, 37–50). If you’re majoring/minoring in mathematics, recall the \(\epsilon\)/\(\delta\) definition of a limit (§1.7) and understand the \(\epsilon\)/\(\delta\) definition in higher dimensions.
- Get good at calculating first and second partial derivatives (9–64). Be able to calculate the equation for a plane tangent to a surface at a point.
- Know how to compute the equation of a plane tangent to a surface at point (1–12). Recall differentials in two-dimensions (§2.9) and know that same idea applies in higher dimensions using a tangent plane.
- Get good at computing derivatives, both explicitly (1–16) and implicitly (25–38).
- Be able to compute gradients (8–12) and directional derivatives (4–7, 13–19, 21–25).
- Know how to compute critical values of a function and determine the location and value of any local extrema or saddle points (1–32), and be able to calculate the global minimum/maximum value of a function on a closed domain (33–40).
- Know how to use the method of Lagrange multipliers to calculate the extreme values of a function subject to some constraints (3–33, 41–53).
- Get good at calculating double integrals over rectangular regions (9–37) and interpreting the volume a double integral corresponds to (38–49).
- Continue getting good at calculating double integrals, but now over regions whose boundaries are the graphs of functions (1–14, 19–40).
- Get good at calculating double integrals in polar coordinates (1–37).
- As applications done of these are really exercises.
- Get good at calculating double integrals that correspond to surface areas (1–14).
- Get good at calculating double integrals and interpreting the volume a double integral corresponds to (1–26).
- Get good at calculating triple integrals in cylindrical coordinates and interpreting the region such a triple integral corresponds to (15–30).
- Get good at calculating triple integrals in spherical coordinates and interpreting the region such a triple integral corresponds to (17–36).
- Understand the algebra of transformations (1–10), get good at computing the Jacobian of a transformation (11–16), and be able to apply a transformation to compute an integral (17–22).
- Be able to sketch a vector field (1–22), and sketch the gradient field of a function (25–34).
- Be able to compute line integrals, regardless of the notation in which they are presented (1–24).
- Determine whether or not a vector field is conservative (3–10), and compute line integrals using the fundamental theorem (1, 2, 11–26).
- Be able to evaluate line integrals using Green’s Theorem (1—18).
- Understand the basics of the \(\operatorname{curl}\) and \(\operatorname{div}\) operators (9–12, 14), and be able to compute them for a given vector field (1–8, 15–20).
- Get used to the notation of parametrically-defined surfaces and how to translate between their algebraic equations and their geometry (1–6, 13–18, 19–26, 33–36), and know how to compute their surface area (39–50).
- Calculate surface integrals of functions (1–19) and flux integrals of vector fields (21–32).
- Be able to wield Stokes’s theorem to calculate surface integrals as line integrals over the boundary of the surface (2–6) as to calculate line integrals as surface integrals of which that curves is a boundary (7–14).
- Be able to wield the Divergence theorem to understand that calculating the divergence of a vector field over a region is the same as calculating the flux of a vector field across the boundary of that region (1–17).