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# Multivariate Calculus Study Guide

## Solutions by Chapter

❶We will also give quite a few definitions and facts that will be useful. On both Part A and B exercises you are expected to write the answer in your own words.

## Description   You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. The 3-D Coordinate System — In this section we will introduce the standard three dimensional coordinate system as well as some common notation and concepts needed to work in three dimensions.

Equations of Lines — In this section we will derive the vector form and parametric form for the equation of lines in three dimensional space. We will also give the symmetric equations of lines in three dimensional space. Note as well that while these forms can also be useful for lines in two dimensional space.

Equations of Planes — In this section we will derive the vector and scalar equation of a plane. We also show how to write the equation of a plane from three points that lie in the plane. Quadric Surfaces — In this section we will be looking at some examples of quadric surfaces. Some examples of quadric surfaces are cones, cylinders, ellipsoids, and elliptic paraboloids. Functions of Several Variables — In this section we will give a quick review of some important topics about functions of several variables.

In particular we will discuss finding the domain of a function of several variables as well as level curves, level surfaces and traces. Vector Functions — In this section we introduce the concept of vector functions concentrating primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well.

We will illustrate how to find the domain of a vector function and how to graph a vector function. We will also show a simple relationship between vector functions and parametric equations that will be very useful at times. Calculus with Vector Functions — In this section here we discuss how to do basic calculus, i. Tangent, Normal and Binormal Vectors — In this section we will define the tangent, normal and binormal vectors. Arc Length with Vector Functions — In this section we will extend the arc length formula we used early in the material to include finding the arc length of a vector function.

Curvature — In this section we give two formulas for computing the curvature i. Velocity and Acceleration — In this section we will revisit a standard application of derivatives, the velocity and acceleration of an object whose position function is given by a vector function.

For the acceleration we give formulas for both the normal acceleration and the tangential acceleration. Cylindrical Coordinates — In this section we will define the cylindrical coordinate system, an alternate coordinate system for the three dimensional coordinate system. As we will see cylindrical coordinates are really nothing more than a very natural extension of polar coordinates into a three dimensional setting.

Spherical Coordinates — In this section we will define the spherical coordinate system, yet another alternate coordinate system for the three dimensional coordinate system. This coordinates system is very useful for dealing with spherical objects.

We will derive formulas to convert between cylindrical coordinates and spherical coordinates as well as between Cartesian and spherical coordinates the more useful of the two. We will also see a fairly quick method that can be used, on occasion, for showing that some limits do not exist. Partial Derivatives — In this section we will the idea of partial derivatives. We will give the formal definition of the partial derivative as well as the standard notations and how to compute them in practice i.

There is only one very important subtlety that you need to always keep in mind while computing partial derivatives. Interpretations of Partial Derivatives — In the section we will take a look at a couple of important interpretations of partial derivatives. First, the always important, rate of change of the function. We will also see that partial derivatives give the slope of tangent lines to the traces of the function. Higher Order Partial Derivatives — In the section we will take a look at higher order partial derivatives.

Unlike Calculus I however, we will have multiple second order derivatives, multiple third order derivatives, etc. Differentials — In this section we extend the idea of differentials we first saw in Calculus I to functions of several variables.

Chain Rule — In the section we extend the idea of the chain rule to functions of several variables. In particular, we will see that there are multiple variants to the chain rule here all depending on how many variables our function is dependent on and how each of those variables can, in turn, be written in terms of different variables. We will also give a nice method for writing down the chain rule for pretty much any situation you might run into when dealing with functions of multiple variables.

In addition, we will derive a very quick way of doing implicit differentiation so we no longer need to go through the process we first did back in Calculus I. In addition to the daily practice problems, at the end of each week a very brief pset will be posted on the homework. A new pset is posted each Friday, and is due the next Wednesday. No late homework will be accepted. Unlike the daily problem lists, the weekly pset if NOT designed to be sufficient practice.

Academic integrity is at the core of our mission as mathematicians and educators, and we take it very seriously. On exams, you may not give or receive help from anyone. Exams in this course are closed book, and no notes, calculators or other electronic devices are permitted. These sessions focus on answering your questions as you work through the homework problems, especially the daily problem lists from the book.

You can get further explanations about the concepts, or ask for help with specific practice problems. Tutorials are open to all Math 13 students.

Past students have found these tutorials to be immensely helpful! Students with disabilities who will be taking this course and may need disability-related classroom accommodations are encouraged to make an appointment to see their instructor as soon as possible. For further information on the available support services, please contact Student Accessibility Services. ## Main Topics

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