## Math 211: Multivariable Calculus Study Guide for Exam 1

The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guideline, and may not explicitly mention everything that you need to study. It is also important you are comfortable with the basics of differentiation and integration (as covered in Calculus I and Calculus II classes).

12.1: You should be able to identify points in three space, and regions defined by inequalities. You should know the distance formula and the equations of cylinders and spheres.

12.2: This section covers the basics of vectors and vector algebra. Make sure you understand them well, including how to find lengths of vectors, unit vectors pointing in a given direction, and scalar multiples and sums of given vectors. You should be able to work with all of these concepts both algebraically and geometrically.

12.3: Know how to find the dot product of two vectors, and how to determine the angle between them using their dot product. You should also know how to find the projection of a vector along another vector, and how to write a vector as a sum of two vectors, one of which is parallel to a given direction and the other perpendicular to that direction.

12.4: Know how to compute the cross product of two vectors and understand its geometric interpretation. You should also know how to use cross products and triple products to find the area of a parallelogram and the volume of a parallelepiped.

12.5: Know how to find the vector and parametric equations of a line given a point and a direction, or two points on the line. You must also know how to find the equation of a plane given a point on the plane and a normal vector, or a line and an additional point contained in the plane, or three points on the plane, or two non-skew lines in the plane. You should also be able to determine the point of intersection of two lines or a line and a plane, and the line of intersection of two planes.

13.1 & 13.2: Make sure you understand what a vector function represents, and know how to find limits, derivatives and integrals involving such functions. You should also be able to find the equation of the tangent line at a point on the space curve traced out by a vector function.

13.3: You must know the formulas to find speed and arc length. You should also know the equations for the unit tangent and the unit normal to a curve at a given point. You do not need to know about curvature or binormal vectors. You will not be required to find the arc length parametrization of a curve.

13.4: You should know what the velocity and acceleration functions mean in terms of derivatives of a vector function. You should also know how to find the tangential and normal components of the acceleration vectors using the methods of section 12.3 (ie by finding the projection of the acceleration onto the velocity). We will not be covering Kepler's Laws.

14.1 & 14.2: Make sure you understand what a function in several variables represents and what level curves (and surfaces) and contour lines are. You should know how to determine limits of several variable functions, or show that it does not exist.

Please be sure to review homework and example problems for the chapters given above! Below is a list of relevant additional practice problems for each section. An "R" in the section number refers to the review exercises at the end of each chapter.

Section Suggested Problems
12.1 15-18, 35-38
12.2 19-25
12.3 1-10, 15-20, 23, 39-44
12.4 1-7, 9-13, 19, 22, 33
12.5 2-14, 19-40, 45-48
12 R 1-12, 15-26
13.1 1-6, 21-26, 27, 40-44
13.2 9-27, 35-42, 47-51
13.3 1-6, 17a-20a
13.4 9-16, 19, 37-42
13 R 1-5, 8, 11ab, 17, 18, 20
14.1 13-22, 32, 34, 36, 43-50, 59-64
14.2 5-10

Solutions for this exam can be found on Moodle.