Vector Calculus Marsden 6th Edition Pdf Download
Geneva Andreotti
I thought, to put it lightly, that the exposition was atrocious and unmotivated, with too much of a focus on memorizing the equations needed to solve problems. I did do well on the calculus questions on the Subject Test [I think], but if I need to teach myself vector calculus again, I can't use Stewart to do it.
I quite liked the way that Leithold covered line integrals. It might be worth taking a look at the rest. In my opinion, better than books, is the course given in MIT, by prof. Denis Auroux, which is available online. I taught myself multivariable calculus using these videos before, and it was most probably the easiest subject I had in university because of that. I really recommend seeing other subjects there too.
This is for good old classical vector analysis, with tensor analysis to boost. It has plenty of solved problems, physical intuition and even proofs for Green, Gauss and Stokes's theorems. It does not contain over 500 hundred problems that you'd find in an usual calculus book, though. They are more carefully selected. It is a Dover title, which is financially a plus.
This one's available for free. It is also a Dover title, which you can buy here. While it does not contain just vector calculus nor is it rigorous, it will build intuition and skill at using these tools in a variety of situations. There are two chapters devoted to Vector Calculus, the second focusing more on the computational part.
However, you can always go back to Apostol's Calculus, volume 2. It should contain everything you need (I haven't personally studied it, but that's the only calculus book I know that could combine rigor and plenty of exercises).
"Vector Analysis" (MATH 4317/5317) is no longer a formal class at ETSU. Such a class existed until the late 1990s. The catalog description for that class was: "Topics in vector algebra, vector functions, scalar, and vector fields; line and surface integrals." This is from the 1989-90 catalog. The prerequisites were Linear Algebra (MATH 2010) and Calculus 3 (MATH 2110).
This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space.
Description: Vector algebra in two and three dimensions, multivariate differential and integral calculus, optimization, vector calculus including the theorems of Green, Gauss, and Stokes. Applications to economics, engineering, and all sciences, with emphasis on numerical and graphical solutions; use of graphing calculators or computers. May not be taken for credit in addition to AMS 261 or MAT 205.
It is impossible to say whether you will be ready to teach multivariable calculus after taking this course. Certainly many of the topics you list will employ some multivariable calculus, but using bits and pieces of an undergrad course as a tool in a higher level course will probably not be the best preparation for teaching.
If you are going to be teaching multivariable calculus, I would recommend getting a few really good textbooks and making sure you can do all of the exercises, motivate all of the definitions and theorems, and feel comfortable enough with the proofs that you can both present the main ideas without any prep work and present a detailed proof with a little bit of prep work. You should also work on having some kind of overarching narrative for the course which makes sense to you.
If you want a "higher level perspective" on multivariable calculus then you should learn some differential geometry. I would recommend John M. Lee's introductions to topological manifolds, smooth manifolds, and riemannian manifolds in that order.
I want to say that this answer is not meant to discourage you from taking the course you mention. It sounds interesting, and it is often true that we really learn something when we are applying it in different situations. I am sure that taking the course will strengthen your understanding of multivariable calculus, just probably not in the most systematic way.
Assuming you already have a good understanding of the material in a standard course in multivariable calculus and vector calculus, I would recommend that you take a class in electricity and magnetism from the physics department. I would guess that a sizable fraction of your students are going to major in engineering and physics, and their courses in electricity and magnetism will likely be where they use vector calculus most heavily. So as their teacher, you ought to understand that material yourself. If you don't already have a lot of physics under your belt, then taking a physics class will broaden your horizons more than studying more analysis will.
3. In many of the problems in this book, once you've done the "20E" (vector calculus) part of the question, you end up with the "20B" part - having to evaluate an integral. This term's course is not meant to be all about doing integrals, but at the same time you mustn't forget how to do them! There are certain types which come up repeatedly in vector calculus (for example integrals of powers of sin and cos, which often appear when using spherical coordinates) and even if you find these frightening to start with, you'll be used to them by the end of the course. Nevertheless, occasionally this book leads you to a genuinely hard integral... if you really can't get the thing to come out by yourself, don't waste too much time on it - just use some software and move on. In my exams I'll make sure that only "reasonable" integrals will be needed.
4. Annoyingly, there are lots of different systems of terminology and notation used in vector calculus - if you look at different books you will find different names and symbols. That means that to read papers and references involving vector calculus you have to be comfortable, in principle, with all of them! Some of the terms and symbols used in this book strike me as confusing, and I won't necessarily use the same ones, but I'll try to maintain somewhere on this webpage a list of these alternatives. You can use whichever conventions you prefer.
In 20C you learned about functions of several variables and their partial derivatives. This course continues directly on from 20C (I have no idea why it's called 20E!) and concerns calculus primarily in three dimensions. The main ideas all originated from the 19th century study of electromagnetism, and the culmination of the course is seeing how to combine various simple experimental observations about EM fields to arrive at Maxwell's equations, the partial differential equations governing EM theory. The language of vector calculus gives us powerful methods for writing and working with these equations in a way that doesn't depend on what coordinate system we use and lets us understand the intrinsic geometrical meaning of the various terms. The same techniques are incredibly useful in all parts of the physical sciences.
Hint: if we move in R^3 from (1,0,1) to the nearby point (1,0,1)+h (for some small vector h), the linear approximation tells us that f changes by Df (evaluated at (1,0,1)) multiplied by h. If we want f to _stay constant_ - that is, h is moving us _along_ (=in a direction tangent to) the surface, we need this change to be 0.
2.3. #28 It's not clear when they say "f: R^n -> R^m is a linear map" whether they mean in the "geometrical sense" (having a flat graph, which means being of the form f(x) = Ax+b, where A is an mxn matrix and b a fixed m-vector) or in the "linear algebra sense" (which means satisfying the laws f(x_1+x_2) = f(x_1) + f(x_2) and f(lambda.x) = lambda.f(x), and means being of the form f(x) = Ax without the b). I think they actually intended the linear algebra sense, but you should be able to work out the derivative in both cases and then see what is special about the second...
This is the sixth edition of a popular advanced calculus text. It has been completely redesigned, yet retains the basic structure and approach of the earlier editions. Part of the appeal of the book has been the balance of its treatment across theory, applications, historical notes and optional material. That remains. What has been added are new examples, more exercises (with a clearer grading of difficulty) and improved artwork with significantly improved three-dimensional figures.
One of the reasons this book has been as successful is probably its focus on the average student. The authors have taken pains throughout to walk students through detailed explanations and worked-out examples, and then test their understanding with more routine exercises before presenting more challenging problems. Their object is simply to take a direct, fairly concrete path to teaching multivariable calculus without many detours. So, for example, instead of introducing the more abstract language of linear transformations the authors describe the derivative in terms of a matrix of partial derivatives.
The four fundamental theorems of vector calculus are generalizationsof the fundamental theorem of calculus. The fundamnetal theorem of calculusequates the integral of the derivative $G'(t)$to the values of $G(t)$ at the interval boundary points:\begingather* \int_a^b G'(t) dt = G(b)-G(a).\endgather*Similarly, the fundamental theorems of vector calculus state that an integralof some type of derivative over some object is equal to the values of functionalong the boundary of that object.
Recognizing the similarity of the four fundamental theorems canhelp you understand and remember them. Here we summarize the theorems andoutline their relationships to the various integrals you learned in multivariable calculus.
Often, we are not given the potential function, but just the integral in terms of a vector field $\dlvf$: $\dlint$. We can use the gradient theorem only when $\dlvf$ isconservative,in which case we can find a potential function$\dlpf$ so that $\nabla \dlpf = \dlvf$. Then,\beginalign* \dlint = \dlpf(\vcq) - \dlpf(\vcp),\endalign*where $\vcp$ and $\vcq$ are the endpoints of $\dlc$.
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