File Name: multivariable calculus problems and solutions .zip
There are four versions of the text in two formats. The book was last updated October 7, , The initial edition is identical to the last updated version.
On the other hand, vector quantity implies the physical quantity which comprises of both magnitude and direction. Vector calculus is the essential mathematical tool for such analysis.
Here are a set of practice problems for the Calculus III notes. Click on the " Solution " link for each problem to go to the page containing the solution. Note that some sections will have more problems than others and some will have more or less of a variety of problems.
Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a listing of sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section. Practice Quick Nav Download. 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 look at 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. Directional Derivatives — In the section we introduce the concept of directional derivatives. With directional derivatives we can now ask how a function is changing if we allow all the independent variables to change rather than holding all but one constant as we had to do with partial derivatives.
In addition, we will define the gradient vector to help with some of the notation and work here. The gradient vector will be very useful in some later sections as well. We will also give a nice fact that will allow us to determine the direction in which a given function is changing the fastest. We will also see how tangent planes can be thought of as a linear approximation to the surface at a given point. Gradient Vector, Tangent Planes and Normal Lines — In this section discuss how the gradient vector can be used to find tangent planes to a much more general function than in the previous section.
We will also define the normal line and discuss how the gradient vector can be used to find the equation of the normal line. Relative Minimums and Maximums — In this section we will define critical points for functions of two variables and discuss a method for determining if they are relative minimums, relative maximums or saddle points i.
Absolute Minimums and Maximums — In this section we will how to find the absolute extrema of a function of two variables when the independent variables are only allowed to come from a region that is bounded i. Double Integrals — In this section we will formally define the double integral as well as giving a quick interpretation of the double integral. Double Integrals over General Regions — In this section we will start evaluating double integrals over general regions, i.
The regions of integration in these cases will be all or portions of disks or rings and so we will also need to convert the original Cartesian limits for these regions into Polar coordinates. Triple Integrals — In this section we will define the triple integral. We will also illustrate quite a few examples of setting up the limits of integration from the three dimensional region of integration.
Getting the limits of integration is often the difficult part of these problems. We will also be converting the original Cartesian limits for these regions into Cylindrical coordinates. We will also be converting the original Cartesian limits for these regions into Spherical coordinates. In this section we will generalize this idea and discuss how we convert integrals in Cartesian coordinates into alternate coordinate systems.
Surface Area — In this section we will show how a double integral can be used to determine the surface area of the portion of a surface that is over a region in two dimensional space. Area and Volume Revisited — In this section we summarize the various area and volume formulas from this chapter. Vector Fields — In this section we introduce the concept of a vector field and give several examples of graphing them. We also revisit the gradient that we first saw a few chapters ago.
Line Integrals — Part I — In this section we will start off with a quick review of parameterizing curves. This is a skill that will be required in a great many of the line integrals we evaluate and so needs to be understood. We will then formally define the first kind of line integral we will be looking at : line integrals with respect to arc length. We also introduce an alternate form of notation for this kind of line integral that will be useful on occasion.
We will also see that this particular kind of line integral is related to special cases of the line integrals with respect to x, y and z. Fundamental Theorem for Line Integrals — In this section we will give the fundamental theorem of calculus for line integrals of vector fields. This will illustrate that certain kinds of line integrals can be very quickly computed.
We will also give quite a few definitions and facts that will be useful. We will also discuss how to find potential functions for conservative vector fields. Curl and Divergence — In this section we will introduce the concepts of the curl and the divergence of a vector field. Parametric Surfaces — In this section we will take a look at the basics of representing a surface with parametric equations.
We will also see how the parameterization of a surface can be used to find a normal vector for the surface which will be very useful in a couple of sections and how the parameterization can be used to find the surface area of a surface. Surface Integrals — In this section we introduce the idea of a surface integral.
With surface integrals we will be integrating over the surface of a solid. In other words, the variables will always be on the surface of the solid and will never come from inside the solid itself. Divergence Theorem — In this section we will discuss the Divergence Theorem.
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you have a sufficient mastery of the subject for multivariable calculus. We first list elizabethsid.org html//handouts/MVT elizabethsid.org Solution: This problem requires the chain rule.
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Good luck! Problem 1. These questions are designed to ensure that you have a su cient mastery of the subject for multivariable calculus. Necessary background is first year calculus through infinite series. Calculus of variations, branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible.
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