5
votes
0answers
31 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
1
vote
0answers
18 views

Existence and Uniqueness of Solutions of PDEs

I have been looking into the Cauchy-Kovalevskaya Theorem where one can "establish the local existence of analytic solutions to a system of PDEs". I wanted to see an application (for example, see ...
2
votes
1answer
35 views

Third Order PDE written as a System of (Linear) First Order PDEs

I need to rewrite the PDE $$f_{y}+ff_{x}+f_{xxx}=0,$$ where $f=f(x,y)$ as a system of first order quasi-linear PDEs. I have no idea how to tackle this problem. Any form of help will be appreciated. ...
1
vote
2answers
74 views

Typed version of Newton's Principia Mathematica

I need a typed pdf version of Newton's Principia. Is it available for free online? And I also need the proof of universal law of gravity and the elliptical orbits of planets(If there's no typed ...
0
votes
0answers
31 views

Reference Request: Fubini's theorem for non-negative functions

I have never seen this (1st page) formulation of Fubini's theorem in the literature. Does anyone know where I can find it? In every calculus book (e.g. Apostol, Courant, etc.) I looked, the authors ...
2
votes
0answers
62 views

Simplifying a Vector Integral

While reading the book - Cercignani, Theory and Applications of Boltzmann Transport Equation (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ ...
0
votes
0answers
56 views

Multivariable calculus and real analysis in one semester. What is the best way to study for such course?

I am in my first year of EECS and planning on taking a lot of maths classes. I have already taken single variable calculus and linear algebra and did well in them and decided to take multivariable ...
1
vote
0answers
32 views

Less Terse alternative to Advanced Calculus by Folland.

I am currently in an advanced calculus class in university. We use Advanced Calculus by Folland. When I try to follow along the book I find that it is not verbose enough, and has too few examples. I ...
0
votes
0answers
17 views

Nature of Hessian of a function of a matrix

If input to a differentiable function is a matrix, what is the nature of Hessian of the function? Is it a tensor or something? This is a simple question, but I guess I am not sure where refer to, to ...
0
votes
1answer
45 views

Best Book or Source for learning Multivariable Calculus!

I urgently need some sources for learning multivariable calculus in an efficient way, without too much intuition. I'd like it to be explained in a clear and concise way and with a number of worked ...
1
vote
0answers
33 views

Best book for learning multiple integrals, line integrals, greens theorem etc..

I've been searching for a book that teaches multiple integrals and such in a way that I can understand, I need to learn it quickly, so I don't need too much of the intuition, I just need to be able to ...
2
votes
1answer
77 views

Learning Advanced Mathematics

I'm a 12th grade student and I've recently developed a passion for mathematics . Currently my knowledge in this particular area is comprised by : single-variable calculus , trigonometry , geometry , ...
1
vote
2answers
121 views

Video lectures and reference book Multivariable calculus

I am in a particular situation that I am doing Master's in a Computer Science related degree, and I would like to take the course on Convex Optimisation which is taught by the Machine Learning ...
5
votes
2answers
303 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem: Let $f(x):\Omega\longrightarrow \mathbb R^n$ a $C^2$ function where $\Omega$ is an open subset of ...
0
votes
1answer
34 views

Fréchet Derivative Bibliography

Good night, I'm new studying the Frechet Derivative. I still don't understand the concept of it. My question is...do you have some bibliography about it you can recommend me? I'd really appreciate ...
1
vote
1answer
53 views

How should I prove $\operatorname{vol}_{n+1}B_{n+1}=\int_0^1 \operatorname{vol}_n S^n(r)dr$ without using spherical coordinates?

Let $B_n:=\{x\in{\Bbb R}^n:|x|\leq 1\}$ and $S^n(r):=\{x\in{\Bbb R}^{n+1}:|x|=r\}$. Then we have the following formula $$ \operatorname{vol}_{n+1}B_{n+1}=\int_0^1 \operatorname{vol}_n S^n(r)dr. ...
2
votes
0answers
84 views

Multivariable calculus along with tensors …etc to start studying General Relativity

I bought Spivak Calculus on Manifolds last time and I was really really disappointed... I opened the first chapters and I understood nothing of what he was saying. But i need to understand ...
0
votes
0answers
43 views

Calculus with leibniz notation

Is there a modern book covering in depth calculus and multivariable calculus (maybe also real analysis?) using only Leibniz ($dx$) notation?
0
votes
1answer
69 views

Advanced Calculus Resource for Multivariate and Complex Calculus

I need a good resource - textbook, online resource, video lecture etc.- that explains the multivariate calculus really well; the topics I want to make clear are the Hessian Matrix concept and the ...
0
votes
0answers
30 views

Proof of double integration.

In the book "A Second Course in Calculus" by Serge Lang, there is a chapter on multiple integrals. The book does not give a justification as to why $$\lim\limits_{\Delta x\to 0,\Delta y\to ...
2
votes
1answer
169 views

Prove that hyperspherical coordinates are a diffeomorphism, derive Jacobian

The explicit form for the transformation into hyperspherical coordinates is $$x_1 = r\sin\theta_1 \sin\theta_2 \dotsb \sin \theta_{n-1} \\ x_2 = r\sin\theta_1 \sin\theta_2 \dotsb \cos \theta_{n-1} \\ ...
0
votes
1answer
110 views

Reference request - second derivative test for function of two variables that includes details of what you can infer when discriminant is zero

The second derivative test for functions of two variables as I have learned and taught in calculus classes says, in part, that if at a point $D=f_{xx}f_{yy}-(f_{xy})^2$ is zero then we can tell ...
0
votes
0answers
47 views

Exercise references

I could recommend any good text analysis, or perhaps a list of exercises with good problems (for show) on dips, submersiones and implicit functions. I appreciate any references.
2
votes
1answer
55 views

What functions on the plane (and on $\mathbb{R}^n$) have projection-valued derivatives?

Thinking about a more general problem I am trying to work out a specific case: If $U\subset \mathbb{R}^2$ is a connected open set what are the differentiable (or $C^r$) functions $f\colon ...
2
votes
0answers
399 views

Book Recommendation - Hard problems for Multivariable Calculus w/ Solutions

I'm looking for a text that covers roughly what's sometimes called "Calculus III" or multivariable calculus.* But this text must satisfy certain additional criteria: (1) It must be more in-depth (and ...
2
votes
0answers
209 views

Cauchy's theorem for integral homotopic closed curve in $G\subset\mathbb{C}^n$.

Recall Cauchy's theorem (third version in the Conway's book "Function of one complex variable", thm 6.7. page 90 in the second edition): Let $f$ be an analytic function on $F\subset\mathbb{C}$ and ...
1
vote
1answer
231 views

Callahan's Advanced Calculus: A Geometric View vs Hubbard's Vector C, L A, and Di Forms vs Ad- Calculus: A Differential Forms Approach by edwards

My friend give me the chance to get one of those book for free,I've learned single variable calculus,which one of these do you think would serve best my purposes?which are learning deeply ...
2
votes
1answer
93 views

Study of Matrix Calculus

I need to study matrix calculus such as integration, differentiation, differentiation of functions of determinants and inverse matrices and then also other matrix based calculations such as ...
2
votes
1answer
44 views

Inverse function theorem: Why is $\frac {\partial \phi }{\partial y} = \frac {-\partial F / \partial y} {\partial F / \partial z}$?

Given $F(x,y,z) = 0$, $\partial F/\partial z \neq 0$ at $p_0$, by the implicit function theorem we can solve for $z=\phi(x,y)$ near $p_0$. I am told that $$\frac {\partial \phi }{\partial y} = \frac ...
4
votes
3answers
277 views

A question on generalization of the concept of derivative

I am looking for some material to understand the process of generalization of the concept of derivative. I would not like to just read and apply the definition of the concept of differentiation in ...
2
votes
1answer
67 views

Show $\partial _x \int_{(x_0, y_0)}^{(x,y)}P(s,t)ds + Q(s,t)dt = P(x,y)$

There is a theorem from advanced calculus that I'm trying to prove. Suppose $P(x,y)$, $Q(x,y) \in C^2$ on a simply connected domain $D$, and suppose that $P_y = Q_x$ (i.e. $\omega = Pdx + Qdy$ is ...
1
vote
1answer
119 views

Second Text for Multivariable Calculus

I took a rather disappointing multivariable calculus course this semester -- the (visiting) professor was not demanding at all. We didn't get to what is in most standard calculus III curriculum. What ...
1
vote
0answers
266 views

What does it mean by piecewise smooth boundary?

I will be highly obliged if anyone can give me any reference where i can get the definition of domain (in $\mathbb{R^n}$) with piecewise smooth boundary. My question is when a domain in ...
2
votes
1answer
65 views

Reference for an integral's convergence on an $n$-ball when $n>2$.

I was searching for a reference of a standard result from calculus. Unfortunately I couldn't find it. I think that's mostly due to I am not familiar with any english calculus book. So I am ...
4
votes
3answers
149 views

Notes about evaluating double and triple integrals

I'm searching notes and exercises about multiple integrals to calculate volume of functions, but the information I find in internet is very bad. Can someone recommend me a book, pdf, videos, ...
2
votes
4answers
228 views

Calculus book for people who know limits

I have the probably slightly unusual background of being quite comfortable with real numbers, functions, limits, sequences, series, etc, but having no knowledge of calculus beyond the definitions of ...
5
votes
0answers
144 views

Algorithm to calculate multiple integral.

One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
3
votes
2answers
281 views

A sequel for Elementary Analysis by Ross?

I've been learning real analysis from this book: Elementary Analysis, K.A. Ross I really liked the style of this book. It is quite old, and sometimes very difficult, but I guess I liked the way it ...
1
vote
1answer
140 views

Divergence Theorem and Mean Curvature as applied to Tension in a Membrane

I'm reading a paper on tension in a membrane and am currently stuck at this part. The paper so far reads: We consider a portion $S^M$ of a membrane $\Omega^M$, where $\hat k$ denotes the unit ...
0
votes
1answer
257 views

Continuity of multivariable functions when “component functions” are continuous

Given topological spaces $X_1, X_2, \dotsc, X_n, Y$, consider a multivariable function $f : \prod_{i = 1}^nX_i \to Y$ such that for any $(x_1, x_2, \dotsc, x_n) \in \prod_{i = 1}^nX_i$, the functions ...
1
vote
0answers
88 views

Reference books or any sources advice for my advanced calculus course

My course outline: Differentiation of functions of several variables: partial derivatives, differential, differentiability, inverse function theorem, implicit function theorem, free extremum ...
1
vote
0answers
46 views

the usual examples for multi-variable integrals

well sadly in our exam our tutor is going to ask us to calculate different integrals etc. Well we're studying maths and not "being a good calculator" but in contrast to give exercises about simple ...
5
votes
4answers
3k views

Multivariable Calculus books similar to “Advanced Calculus of Several Variables” by C.H. Edwards

I am currently trying to teach myself multivariable calculus using C.H. Edwards' "Advanced Calculus of Several Variables", but the text unfortunately doesn't have very many problems with solutions. ...
2
votes
1answer
288 views

Calculus 3 Explained

I am trying to learn some calculus 3 and I understand HOW to do the problems but I just don't understand WHY I'm doing what I'm doing. So does anyone have any good recommendations on books that are ...
3
votes
0answers
784 views

Good introductory book for matrix calculus

Hi I am an electronics graduate and working on image processing for the past one year...I have a basic exposure to linear algebra(thanks to Gilbert Strang..!!!). Now I am facing problems with matrix ...
1
vote
0answers
194 views

A rigorous book (or preferrably set of notes) on classic multivariable calculus-analysis?

This is different to (Theoretical) Multivariable Calculus Textbooks as I want a classical treatment of line and surface integrals without the notion of a differential form. Prerequisites: Paths, ...
0
votes
1answer
57 views

Practise with Smooth Functions and Manifolds

I'm trying to get an intuition for smooth manifolds, and in particular the smoothness of transition functions. I haven't done that much calculus on $\mathbb{R}^n$ before, and would like to practice ...
0
votes
2answers
262 views

Multivariable Product Rule, Integration by Parts, Derivative, etc.

I am searching for a book on multi-variable calculus that explains multi-variable product, multi-variable integration by parts, etc. As an example, here's a simple problem that I would like to be ...
2
votes
1answer
770 views

about a good book - Vector Calculus[by Jerold E. Marsden, Anthony J. Tromba ]

I start reading Vector Calculus by Jerold E. Marsden, Anthony J. Tromba and I want to know if there is a book with the answers of the exercises. I like a lot this book, it seems to be made for a ...
0
votes
2answers
159 views

Free PDF for MV Calculus

I was looking for a free PDF from which I can review MV calculus. Specifically: MV Limits, Continuity, Differentiation. Differentiation of vector and scalar fields Surface/Multiple Integrals A ...