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Vector Calculus

Vector Calculus


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Vector calculus is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus. The modules in this section of the core complement Corral's Vector Calculus TextMap and the Vector Calculus (UCD Mat 21D) Libretext. Check there for more vector calculus content.


Vector Calculus in a Nutshell

The most important object in our course is the vector field, which assigns a vector to every point in some subset of space.

We'll cover the essential calculus of such vector functions, and explore how to use them to solve problems in partial differential equations , wave mechanics, electricity and magnetism, and much more!

This quiz kicks off a short intro to the essential ideas of vector calculus.

A partial differential equation (pde for short) is an equation involving unknown multivariable functions and their partial derivatives.


Contents

Vectors in three dimensions Edit

In 3d Euclidean space, R > 3 , the standard basis is ex, ey, ez. Each basis vector points along the x-, y-, and z-axes, and the vectors are all unit vectors (or normalized), so the basis is orthonormal.

Throughout, when referring to Cartesian coordinates in three dimensions, a right-handed system is assumed and this is much more common than a left-handed system in practice, see orientation (vector space) for details.

For Cartesian tensors of order 1, a Cartesian vector a can be written algebraically as a linear combination of the basis vectors ex, ey, ez:

where the coordinates of the vector with respect to the Cartesian basis are denoted ax, ay, az. It is common and helpful to display the basis vectors as column vectors

when we have a coordinate vector in a column vector representation:

A row vector representation is also legitimate, although in the context of general curvilinear coordinate systems the row and column vector representations are used separately for specific reasons – see Einstein notation and covariance and contravariance of vectors for why.

The term "component" of a vector is ambiguous: it could refer to:

  • a specific coordinate of the vector such as az (a scalar), and similarly for x and y, or
  • the coordinate scalar-multiplying the corresponding basis vector, in which case the "y-component" of a is ayey (a vector), and similarly for x and z.

A more general notation is tensor index notation, which has the flexibility of numerical values rather than fixed coordinate labels. The Cartesian labels are replaced by tensor indices in the basis vectors exe1, eye2, eze3 and coordinates axa1, aya2, aza3. In general, the notation e1, e2, e3 refers to any basis, and a1, a2, a3 refers to the corresponding coordinate system although here they are restricted to the Cartesian system. Then:

It is standard to use the Einstein notation—the summation sign for summation over an index that is present exactly twice within a term may be suppressed for notational conciseness:

An advantage of the index notation over coordinate-specific notations is the independence of the dimension of the underlying vector space, i.e. the same expression on the right hand side takes the same form in higher dimensions (see below). Previously, the Cartesian labels x, y, z were just labels and not indices. (It is informal to say "i = x, y, z").

Second order tensors in three dimensions Edit

A dyadic tensor T is an order 2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = ab. Analogous to vectors, it can be written as a linear combination of the tensor basis exexexx , exeyexy , . ezezezz (the right hand side of each identity is only an abbreviation, nothing more):

Representing each basis tensor as a matrix:

then T can be represented more systematically as a matrix:

See matrix multiplication for the notational correspondence between matrices and the dot and tensor products.

More generally, whether or not T is a tensor product of two vectors, it is always a linear combination of the basis tensors with coordinates Txx, Txy, . Tzz:

while in terms of tensor indices:

Second order tensors occur naturally in physics and engineering when physical quantities have directional dependence in the system, often in a "stimulus-response" way. This can be mathematically seen through one aspect of tensors - they are multilinear functions. A second order tensor T which takes in a vector u of some magnitude and direction will return a vector v of a different magnitude and in a different direction to u, in general. The notation used for functions in mathematical analysis leads us to write v - T(u) , [1] while the same idea can be expressed in matrix and index notations [2] (including the summation convention), respectively:

By "linear", if u = ρr + σs for two scalars ρ and σ and vectors r and s, then in function and index notations:

and similarly for the matrix notation. The function, matrix, and index notations all mean the same thing. The matrix forms provide a clear display of the components, while the index form allows easier tensor-algebraic manipulation of the formulae in a compact manner. Both provide the physical interpretation of directions vectors have one direction, while second order tensors connect two directions together. One can associate a tensor index or coordinate label with a basis vector direction.

The use of second order tensors are the minimum to describe changes in magnitudes and directions of vectors, as the dot product of two vectors is always a scalar, while the cross product of two vectors is always a pseudovector perpendicular to the plane defined by the vectors, so these products of vectors alone cannot obtain a new vector of any magnitude in any direction. (See also below for more on the dot and cross products). The tensor product of two vectors is a second order tensor, although this has no obvious directional interpretation by itself.

The previous idea can be continued: if T takes in two vectors p and q, it will return a scalar r. In function notation we write r = T(p, q), while in matrix and index notations (including the summation convention) respectively:

The tensor T is linear in both input vectors. When vectors and tensors are written without reference to components, and indices are not used, sometimes a dot ⋅ is placed where summations over indices (known as tensor contractions) are taken. For the above cases: [1] [2]

motivated by the dot product notation:

More generally, a tensor of order m which takes in n vectors (where n is between 0 and m inclusive) will return a tensor of order mn , see Tensor § As multilinear maps for further generalizations and details. The concepts above also apply to pseudovectors in the same way as for vectors. The vectors and tensors themselves can vary within throughout space, in which case we have vector fields and tensor fields, and can also depend on time.

Following are some examples:

An applied or given. . to a material or object of. . results in. . in the material or object, given by:
unit vector n Cauchy stress tensor σ a traction force t t = σ ⋅ n =<oldsymbol >cdot mathbf >
angular velocity ω moment of inertia I an angular momentum J J = I ⋅ ω =mathbf cdot <oldsymbol >>
moment of inertia I a rotational kinetic energy T T = 1 2 ω ⋅ I ⋅ ω <2>><oldsymbol >cdot mathbf cdot <oldsymbol >>
electric field E electrical conductivity σ a current density flow J J = σ ⋅ E =<oldsymbol >cdot mathbf >
polarizability α (related to the permittivity ε and electric susceptibility χE) an induced polarization field P P = α ⋅ E =<oldsymbol >cdot mathbf >
magnetic H field magnetic permeability μ a magnetic B field B = μ ⋅ H =<oldsymbol >cdot mathbf >

For the electrical conduction example, the index and matrix notations would be:


Three vector calculus operations which find many applications in physics are:

1. The divergence of a vector function
2. The curl of a vector function
3. The Gradient of a scalar function
These examples of vector calculus operations are expressed in Cartesian coordinates, but they can be expressed in terms of any orthogonal coordinate system, aiding in the solution of physical problems which have other than rectangular symmetries. Index


Vector Calculus

This is a fairly short chapter. We will be taking a brief look at vectors and some of their properties. We will need some of this material in the next chapter and those of you heading on towards Calculus III will use a fair amount of this there as well.

Here is a list of topics in this chapter.

Basic Concepts – In this section we will introduce some common notation for vectors as well as some of the basic concepts about vectors such as the magnitude of a vector and unit vectors. We also illustrate how to find a vector from its starting and end points.

Vector Arithmetic – In this section we will discuss the mathematical and geometric interpretation of the sum and difference of two vectors. We also define and give a geometric interpretation for scalar multiplication. We also give some of the basic properties of vector arithmetic and introduce the common (i), (j), (k) notation for vectors.

Dot Product – In this section we will define the dot product of two vectors. We give some of the basic properties of dot products and define orthogonal vectors and show how to use the dot product to determine if two vectors are orthogonal. We also discuss finding vector projections and direction cosines in this section.

Cross Product – In this section we define the cross product of two vectors and give some of the basic facts and properties of cross products.


Vector Calculus

This is the homepage for the free book Vector Calculus, by Michael Corral (Schoolcraft College).

Download

Java code samples from the book: calc3book_java.zip
MATLAB/Octave versions: ParallelizationArea.zip (courtesy of Prof. Benson Muite (University of Michigan))
Sage versions: calc3book_sage.zip

Note: The PDF was built using TeXLive 2011 and Ghostscript 9.53 under Linux (Fedora).
LaTeX source code: calc3book-1.0-src.tar.gz

The book is distributed under the terms of the GNU Free Documentation License, Version 1.2.

Buy at Lulu.com

You can buy a printed and bound paperback version of the book with grayscale graphics for $10 plus shipping at Lulu.com here.

Book Description

This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: basic vector algebra lines, planes and surfaces vector-valued functions functions of 2 or 3 variables partial derivatives optimization multiple integrals line and surface integrals.

The book also includes discussion of numerical methods: Newton's method for optimization, and the Monte Carlo method for evaluating multiple integrals. There is a section dealing with applications to probability. Appendices include a proof of the right-hand rule for the cross product, and a short tutorial on using Gnuplot for graphing functions of 2 variables.

There are 420 exercises in the book. Answers to selected exercises are included.

Table of Contents

  1. Vectors in Euclidean Space
    • Introduction
    • Vector Algebra
    • Dot Product
    • Cross Product
    • Lines and Planes
    • Surfaces
    • Curvilinear Coordinates
    • Vector-Valued Functions
    • Arc Length
  2. Functions of Several Variables
    • Functions of Two or Three Variables
    • Partial Derivatives
    • Tangent Plane to a Surface
    • Directional Derivatives and the Gradient
    • Maxima and Minima
    • Unconstrained Optimization: Numerical Methods
    • Constrained Optimization: Lagrange Multipliers
  3. Multiple Integrals
    • Double Integrals
    • Double Integrals Over a General Region
    • Triple Integrals
    • Numerical Approximation of Multiple Integrals
    • Change of Variables in Multiple Integrals
    • Application: Center of Mass
    • Application: Probability and Expected Value
  4. Line and Surface Integrals
    • Line Integrals
    • Properties of Line Integrals
    • Green's Theorem
    • Surface Integrals and the Divergence Theorem
    • Stokes' Theorem
    • Gradient, Divergence, Curl and Laplacian
  • Bibliography
  • Appendix A: Answers and Hints to Selected Exercises
  • Appendix B: Proof of the Right-Hand Rule for the Cross Product
  • Appendix C: 3D Graphing with Gnuplot

(2021-01-05) Cleaned up the web page to make it less hideous and more consistent with the revamped page for Elementary Calculus.

  • Appendix A: The answer to Exercise 5 from Section 1.9 is now fixed.
  • Section 1.1: In Example 1.3(d) R^3 now has the correct dimension.
  • Section 1.7: In Example 1.33 a minus is now a plus.
  • Appendix A: The margin overrun in the answers for Section 2.4 has been removed.

Problems with my TeXLive 2014 setup had caused numerous issues when trying to compile the book. I ended up going back to TeXLive 2011 to fix all that, and it worked. So now, after many requests, I have finally restored the ability to buy a printed and bound paperback version on Lulu.com. It's even a buck cheaper than before. See the link near the top of this page.

On a side note, there are many things about the book that I would change now, after the experience of writing the Trigonometry book and especially the Elementary Calculus book, both content-wise and stylistically. I haven't decided on that yet, but if I do re-write Vector Calculus then I would keep the current version available in addition to the new version. Any decision on that wouldn't be for at least another year, though.

(2013-05-21) I finally(!) got around to uploading the MATLAB/Octave versions of the programs in the book, which Prof. Benson Muite (Univeristy of Michigan) kindly sent me over a year ago. I apologize for the delay my only excuse is that my schedule became incredibly hectic over the last year. Now that things have settled down again, I should have some time to start working on a French translation of Vector Calculus (as well as finish Elementary Calculus). There have been many offers from people around the world to translate my books into other languages. For example, Prof. Koichiro Yamashita will post his Japanese translation at http://kymst.net after he finishes it.

The latest version of Vector Calculus contains a correction of a typo in one of the plots (Fig. 1.8.3 on p.54), which Prof. Yamashita found.

(2012-02-13) I ported the Java code examples in Sections 2.6 and 3.4 to Sage, a powerful and free open-source mathematics software system that is gaining in popularity. The Sage code examples are in the calc3book_sage.zip file, and can be run either on the command-line or as worksheets in a Sage notebook. See the included README file for more details.

The reason for doing this is because I received a request a few years ago to rewrite those code examples for Sage. I wasn't as familiar with Sage as I am now, so I finally got around to doing it. In general, I will be using Sage more, and in particular it will be used extensively for the code examples in my upcoming Elementary Calculus book.

(2011-06-29) The latest version of the book is out. The content of the book is basically the same as before. The big change was in switching the math font from txfonts to Fourier-GUTenberg. This was done to make the fonts more consistent. In particular, the fouriernc package makes use of the New Century Schoolbook normal text fonts for numbers and letters in math mode. This way there is no longer the incongruity of having txfonts' Times Roman-like numbers and letters in math mode versus New Century Schoolbook's different-looking numbers and letters in the main text. This change required a bit of space adjustment throughout the text, since some of the symbols in the Fourier-GUTenberg fonts are slightly smaller than those in txfonts. The sans serif font was also changed, from Avant Garde to Helvetica.

Another change was cleaning up the graphics, which also had a mish-mash of inconsistent fonts and other issues (in particular the graphics created with MetaPost and Gnuplot). The Gnuplot graphics were slightly improved over some of the default settings which I had used originally.

These changes in the appearance make the book look better overall, in my opinion, and were long overdue. It also brings the book in line with the general look and feel of my Trigonometry book and my forthcoming Elementary Calculus book (the prequel to this book).

As far as the content in the main text itself, the only changes are:

  • I added a tiny clarification on the relationship between determinants and volumes of parallelepipeds, right before Theorem 1.17 in Section 1.4. In particular I give the conditions for when the determinant gives the positive volume or the negative volume.
  • In Section 1.7 I added a footnote about the left-handedness of the usual definition of the spherical coordinate system used by mathematicians. I did this because physics students may get confused when they see the definitions of &theta and &phi switched in their physics classes.
  • Improved the code listings to use a monospaced font (Bitstream Vera Sans Mono). I still do not know what possessed me to use a proportional font originally.
  • Updated the URL for downloading Java (since Oracle bought Sun Microsystems).
  • Four corrections in the answers in Appendix A: 1.5 #1, 1.9 #3, 4.1 #11, 4.5 #4 (thanks to P. Taskas and G. Strzalkowski).
  • Updated instructions for using Gnuplot in Appendix C (in the Windows version some of the defaults and procedures changed slightly).

Update (2011-06-30): The printed bound version of the book on the Lulu.com site has also been updated with the latest changes.

(2011-04-17) I've written up a very short (10-page) mini-tutorial on using the LaTeX typesetting system. You can download it here: latex-tutorial.pdf
The source code for the tutorial is available here: latex_tutorial.zip
The tutorial was originally created for students in a class I'm teaching this semester, and I've expanded it a bit since then. I hope others find it useful.

(2010-06-06) Typos in the proof of Theorem 1.20(f) on p.53 have been corrected (thanks to F. Dockhorn for finding those). A newer version of the TikZ/PGF graphics package broke the diagrams on pp.60-61, so the code for those diagrams has been updated. Also, I am still working on the prequel - Elementary Calculus - which (barring a miraculous increase in my productivity) will likely not be ready until sometime next year.

(2009-09-13) The author's new book, Trigonometry, is now available. The homepage is located here: mecmath.net/trig/

(2009-07-22) The prequel to this book, which will be titled Elementary Calculus, is in preparation. It will cover calculus of a single variable. The aim is to have it available by the end of this year or early next year. Another book, on trigonometry, is almost finished and should be available here by the end of August 2009. Both books will be free and released under the GNU Free Documentation License, complete with the LaTeX source code.

(2009-07-22) In Appendix A, the answer to Exercise 5 in Section 2.3 was corrected. Thanks to E. Cavazos for pointing out the error. This is the only change in the new version (2009-07-22) from the previous version (2009-03-29).

(2009-07-10) Prof. Marshall Hampton of the University of Minnesota, Duluth has kindly posted some notes on compiling the LaTeX source code for the book under OS X, which you can read here.

Contact

The author of the book, Michael Corral, can be reached via email at


Vector Calculus

What makes this book different?

This text covers most of the standard topics in multivariate calculus and a substantial part of a standard first course in linear algebra. If, in addition, one teaches the proofs in Appendix A, the book can be used as a textbook for a course in analysis.The organization and selection of material differs from the standard approach in four ways.

    We integrate linear algebra and multivariate calculus.

See what students and professors have to say about Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. See the table of contents in pdf form (requires Acrobat Reader).

Instructors' Solution Manual A revised version of the instructor's solution manual, incorporating corrections and some new solutions, is now available from Prentice Hall .

If your are taking your first serious math course and don't have the book, you may find several brief excerpts useful they are from Chapter 0: Preliminaries. They are in pdf format and can be read using Acrobat Reader.

The first discusses how to read mathematics. (Many students who never read their math texts in high school discover this doesn't work in college.)

The second explains why ``all eleven-legged alligators are orange with blue stripes'' is a true statement, as is ``all eleven-legged alligators are black with white stripes.''

The third gives the vocabulary of set theory (and a nice picture of Bertrand Russell shaving his image in the mirror.)


  • Sections and level sets
  • Graphing scalar valued functions
  • Limits
  • Continuity
  • Directional derivatives
  • Partial derivatives
  • Total differentials
  • The Derivative Matrix
  • Tangent planes
  • Linear approximation
  • Differentiability
  • The Simple Chain Rule
  • The Matrix Chain Rule
  • The Paths Chain Rule
  • The gradient
  • Iterated partial derivatives
  1. Look at all the critical points x where the gradient of f(x) = 0. Throw out any that aren't in D .
  2. Look at any points where f isn't differentiable.
  3. Look at the boundary of D . You can this either by
    • Parameterizing the boundary so that you have an unconstrained max/min problem, or by
    • Lagrange multipliers: if the boundary of D is described as a level set by g(x) = 0, then the critical points of f constrained to be on the boundary of D can be found by solving

Vector Calculus Sessions

The Vector Calculus Review Sessions are a review of topics taught in UC San Diego's Vector Calculus course (Math 20E) that you (may) have learned at your previous institution. The session materials have been reviewed by a Math Department Graduate Student and a Faculty member. 

Please note that these sessions do not replace a Vector Calculus course and will not be teaching you the topics. If at any point in the sessions you discover a topic that you have not learned previously in your Vector Calculus course, you are encouraged to enroll in Math 20E at UC San Diego.  There is a Summer Session 2 Vector Calculus (Math 20E) course, that will begin on Monday, August 1st to September 10th (5-weeks).

  • To  support transfer students to review key topics in preparation for attempting the UC San Diego MATH 20E Fulfillment Exam

Program Details:

  • Dates: July 19th - 30th
  • Sign-up Deadline: July 15th
  • Cost: No cost to participate
  • Format: In-person and Zoom options available
  • Content: Review core calculus topics, including topics and problems based on vector calculus
  • Time commitment: Three sessions per week for two weeks, totaling six 1.5 sessions, plus additional independent study

Eligibility

  • Incoming  transfer students
  • Enrolled in a major that requires MATH 20E
  • Have taken a vector calculus course at community college
  • Planning to attempt the MATH 20E fulfillment exam to fulfill the MATH 20E requirement

Topics to be Reviewed:

  1. Double integral over a region and changing the order of integration, mean value inequality
  2. Triple integral
  3. Geometry of Maps from R2 to R2
  4. The change of variables theorem
  5. Vector field
  6. Path Integral, Line Integral
  7. Parametrized Surface and use integral to find the area of a surface
  8. Integrals of Scalar Functions Over Surfaces
  9. Surface Integrals of Vector Field
  10. Green's Theorem
  11. Stoke's Theorem
  12. Gauss's Theorem (Divergence Theorem)
  13. Conservative Fields

Please read: 

If you have not yet taken vector calculus, we do not recommend that you participate in these Topic Review Sessions or attempt the MATH 20E Fulfillment Exam. Instead, we highly encourage you to enroll in MATH 20E within your first year. Consider getting a head start by taking an accelerated 5-week version of the course during Summer Session II, which takes place from August 3rd - September 4th. The deadline to enroll in Summer Session II is July 26th.

If you have questions, please email [email protected]


Vector Calculus

Vector calculus is one of the most useful branches of mathematics for game development. You could say it is the most important if you're willing to play it slightly fast and loose with definitions and include in it the subset of low-dimensional linear algebra that vector calculus relies on for a lot of its computation. Certainly for physics and any advanced graphics, it's vitally important. It's also beautiful and cool, and a lot of fun to work with and apply to solving problems.

For my purposes, vector calculus is the study of how scalars and vectors change in space and time, which sounds a lot like video games to me.

Eventually I will collect more of my writing and thoughts on vector calculus here, including the most useful techniques for solving problems from my experience. In the meantime.

Lecture

At the 2005 Game Developers Conference, I gave a lecture originally titled Why You Should Have Paid Attention in Vector Calculus, and then retitled Neat Stuff from Vector Calculus & Related Subjects. I think this was one of my weaker lectures of the past 10 years because I was really destroyed by the Indie Game Jam that year, and I was incredibly rushed creating the slides, but there's still a bunch of interesting stuff in the presentation. Highlights include:


Watch the video: Calculus 3 - Intro To Vectors (June 2022).


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