Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

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10
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3answers
4k views

Understanding the concept behind the Lagrangian multiplier

I've been trying to understand the principles behind the Lagrangian multipliers and I think I've got a rough understanding of it. Would appreciate it if you guys could help me answer a few questions! ...
24
votes
6answers
2k views

Geometric interpretation of mixed partial derivatives?

I'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. ...
11
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6answers
9k views

Why do we need vectors and who invented it?

It is natural to understand the need for scalars (numbers), but why did we invent vectors? Who invented it and for what? EDIT: As George Lowther pointed out, the problem is too broad; I added the ...
8
votes
5answers
4k views

Need Help: Any good textbook in undergrad multi-variable analysis/calculus?

This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: -Differentiability. -Open mapping theorem. -...
11
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5answers
7k 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. I'...
10
votes
1answer
196 views

dropping injectivity from multivariable change of variables

The change of variables for multivariable integration in Euclidean space is almost always stated for a $C^1$ diffeomorphism $\phi$, giving the familiar equation (for continuous $f$, say) $$\boxed{\...
6
votes
1answer
814 views

On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is ...
3
votes
2answers
3k views

On the absolute value of Jacobian determinant - variable transformation in multi-integral

I would like to change some variables in a integral and encountered to an issue. I create here 2 simple examples to describe my questions: Exp 1. Suppose we want to change $(x,y)$ to $(u,v)$ such ...
9
votes
2answers
1k views

Pushforward of Lie Bracket

I am trying to figure out why the following equality is true : $$f_*[X,Y]=[f_*X,f_*Y]$$ where $f:M\rightarrow N$ is a diffeomorphism, $M$, $N$ are smooth manifolds, $X$, $Y$ are smooth vector fields ...
6
votes
5answers
2k views

Reference for multivariable calculus

I'm looking for a book to learn multivariable calculus that is rigorous, but not overly technical, and also provides meaningful insight. Standard calculus texts like Stewart and Thomas are too sketchy....
6
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0answers
224 views

Multivariable Integral, How to compute it?

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm ...
5
votes
4answers
2k views

What exactly are the “higher order terms” (H.O.T.) in Taylor series expansion in multivariable case?

I know the Taylor series expansion in single variable case: $$ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{3!}f^{(3)}(x_0)(x-x_0)^3 + \frac{1}{4!}f^{(4)}(x_0)(x-x_0)^4 +...
8
votes
3answers
2k views

Can “being differentiable” imply “having continuous partial derivatives”?

Consider the following theorem: Let $E$ be a subset of ${\bf R}^n$, $f:E\to {\bf R}^m$ be a function, $F$ be a subset of $E$, and $x_0$ be an interior point of $F$. If all the partial derivatives $\...
6
votes
2answers
698 views

Why is arc length not a differential form?

I read that the arc length is not a differential form. But I don't understand why it isn't. I understand that differential forms are integrands and arc length is an expression which is integrable. ...
5
votes
1answer
408 views

If $f$ is twice differentiable, $(f(y) - f(x))/(y-x)$ is is differentiable

Suppose $f: \mathbb{R} \to \mathbb{R}$ is a $C^{1}$ function. Then, define a new function $F: \mathbb{R}^{2} \to \mathbb{R}$ by: $$ F(x,y) = \begin{cases} \displaystyle \frac{f(y) - f(x)}{y - x} &...
4
votes
3answers
465 views

correcting a mistake in Spivak

Spivak's Calculus on Manifolds asks the reader to prove this (problem 1-8, pp.4-5): If there is a basis $x_1, x_2, ..., x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \lambda_2, ..., \lambda_n$ ...
2
votes
1answer
9k views

How to tell if a limit of a multi-variable function exists?

Since I began studying limits of multi-variable functions, I have been baffled with this question: how can one tells if a limit exists or not? I don't know if it's the right way to solve this kind of ...
12
votes
5answers
5k views

Del. $\partial, \delta, \nabla $: Correct enunciation

I've come across various different symbols being pronounced as "del". What is the internationally accepted del? If not internationally, then what's the English/American(specify which one if they are ...
11
votes
3answers
395 views

Determine whether $\lim_{(x,y)\to (2,-2)} \dfrac{\sin(x+y)}{x+y}$ exists.

I am trying to determine whether $\lim_{(x,y)\to (2,-2)} \dfrac{\sin(x+y)}{x+y}$ exists. I should be able to use the following definition for a limit of a function of two variables: Let $f$ be a ...
11
votes
1answer
747 views

Problem 3-38 in Spivak´s Calculus on Manifolds

This is not homework. Problem 3-38 reads: Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for $...
9
votes
5answers
1k views

Distinction between vectors and points

I've been wondering for some time now about the difference between a point and a vector. In high school, it was very important to distinguish them from each other, and we used the notation $(x,y,z)$ ...
8
votes
1answer
1k views

Taylor's theorem in Banach spaces

Let $f$ be a real function of a single real variable. Suppose that $f$ is $n$ times differentiable at some $x$, for some integer $n\geq 1$. Making no further assumptions, we have $$ f(x+h) = f(x) + f&...
8
votes
3answers
1k views

Newton's method in higher dimensions explained

I'm studying about Newton's method and I get the single dimension case perfectly, but the multidimensional version makes me ask question... In Wikipedia Newton's method in higher dimensions is ...
7
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3answers
1k views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
7
votes
1answer
163 views

$ \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 $

Evaluate $$I= \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4$$ Answer Options: $1$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$ I need some ...
6
votes
1answer
74 views

If a separately continuous function $f : [0,1]^2 \to \mathbb{R}$ vanishes on a dense set, must it vanish on the whole set?

Assume $f(x,y)$ is defined on $D=[0,1]\times[0,1]$, and $f(x,y)$ is continuous of each separate variables(i.e. if we fix $y$ to $y_0$, then $f(x,y_0)$ is continuous and vice versa). If $f(x,y)$ ...
6
votes
2answers
2k views

Equivalent condition for differentiability on partial derivatives

I want to extend the concept of derivative of a real function of real variable to a function $f:A\subset \mathbb{R}^n \to \mathbb{R}^m$ with $A$ open. If $x_0 \in A$ then I say that $f$ has derivative ...
5
votes
1answer
580 views

Use Stokes's Theorem to show $\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2$

I am a little stuck on the following problem: Use Stokes's Theorem to show that $$\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2,$$ where $C$ is the suitably oriented intersection of the ...
5
votes
1answer
1k views

if the curvature is constant and positive, then it is on the circunference

I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the ...
5
votes
1answer
2k views

Proof that continuous partial derivatives implies differentiability

This is the statement of Theorem 2.8 from Spivak's Calculus on Manifolds. I'd like feedback on if this looks fine as far as a generalization to his proof goes: Theorem: If $f: \mathbb{R}^n \...
4
votes
1answer
244 views

Limit with integral or is this function continuous?

Hello I need to show one identity and one limit. I am having problems with it. notation: $x_i$ is i-th coordinate of $x$ $B(x,r)$ ball with center $x$ and radius $r$ $S(x,r)$ sphere with center $x$...
4
votes
2answers
144 views

Is this vector-valued map Hölder-continuous?

Pick $0<q<1$ and consider the map from $\mathbb{R}^n$ to $\mathbb{R}^n$ that sends $x$ to $|x|^{q-1}x$. Is this map Hölder-continuous (I guess with exponent $\leq q$)? In dimension one, I can ...
1
vote
3answers
127 views

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? I've been attempting this with Lagrange multipliers in a few different ways. However, the ...
11
votes
4answers
2k views

Is the max of two differentiable functions differentiable?

Given that $f$ and $g$ are two real functions and both are differentiable, is it true to say that $h=\max{(f,g)} $ is differentiable too? Thanks
8
votes
1answer
640 views

A limit and a coordinate trigonometric transformation of the interior points of a square into the interior points of a triangle

The coordinate transformation (due to Beukers, Calabi and Kolk) $$x=\frac{\sin u}{\cos v}$$ $$y=\frac{\sin v}{\cos u}$$ transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the ...
8
votes
2answers
410 views

Multivariable limit with logarithm

I have to prove that the limit $$\lim_ {{(x,y)} \to {(0,0)}} \frac{xy^2\ln\frac{|x|}{|y|}}{{(x^2+y^2)}^{\frac 12}}$$ does not exist. I've tried to find two different paths that show that the limit ...
6
votes
3answers
772 views

Proving that the function $\frac{x^2y}{x^2 + y^2}$ is continuous at $(0,0)$.

How would you prove or disprove that the function given by $$f(x,y) = \begin{cases} \frac{x^2y}{x^2 + y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$ is continuous at $(0,0$)?
6
votes
2answers
721 views

A little help integrating this torus?

Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by $$\mathbf{F}(x,y,z)=(x,y,z).$$ Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus given ...
6
votes
2answers
262 views

Is this a correct use of the squeeze theorem?

I have to find the following limit: $$ \lim_{x\to0,y\to0} \frac{x^2y^2}{x^2+y^4}=[\frac{0}{0}] $$ I try to reach the origin moving on the y-axis ($x=0$): $$ \lim_{y\to0} \frac{0}{y^4}=0 $$ I get ...
5
votes
2answers
991 views

Is the boundary of the unit sphere in every normed vector space compact?

I wanted to ask whether the boundary of the unit sphere in every normed vector space is compact? I know that this is true for simple examples, but how is it in general?
5
votes
5answers
2k views

What is the intuition behind the unit normal vector being the derivative of the unit tangent vector?

I've seen the math, but... It just doesn't make sense to me. How is the slope going to point perpendicular to the vector that is clearly a straight line not going in that direction?
5
votes
1answer
12k views

Expressing the determinant of a sum of two matrices?

Can $$\det(A + B)$$ be expressed in terms of $$\det(A), \det(B), n$$ where $A,B$ are $n\times n$ matrices? # I made the edit to allow $n$ to be factored in.
5
votes
5answers
375 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
4
votes
2answers
65 views

Single variable complex analysis vs the world of the functions $f:\Bbb R^2 \to \Bbb R^2$.

Is there any advantage to studying single variable complex analysis as it is right now instead of just studying the world of the functions $f:\Bbb R^2 \to \Bbb R^2$? I'm asking this because any ...
4
votes
3answers
192 views

Why spherical coordinates is not a covering?

Maybe this is an idiot question and I'm committing a trivial mistake. Let $\phi (\theta, \varphi) = (\cos \theta \sin \varphi, \sin \theta\sin \varphi, \cos \varphi)$ be the usual covering of the ...
4
votes
2answers
366 views

Understanding the derivative as a linear transformation

It's been a while now I am studying multivariable calculus and the concept of differentiation in space (or higher dimension). I saw relative posts but one question remains. I can't understand the ...
3
votes
1answer
2k views

Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1

The task is as follows: Given: $$F(x,y,z) = 0$$ Goal: Show $\frac{\partial z}{\partial y}|_x \frac{\partial y}{\partial x}|_z \frac{\partial x}{\partial z} |_y = -1$ Here is my work so ...
3
votes
1answer
979 views

Showing something isn't a manifold

So I'm following some notes that are introducing manifolds with pretty minimal prerequisites. What I want to do is show where the image of $\phi: \mathbb{R}\rightarrow \mathbb{R^2}$ $t\mapsto (t-\sin(...
3
votes
1answer
86 views

What's the name of these two surfaces?

I've plot two implicit surfaces which are shown in the above, I only know their expression, but I don't know how to call them.
3
votes
2answers
104 views

Use implicit function theorem to show $O(n)$ is a manifold

In class today our teacher mentioned that one can use the implicit function theorem to show that $O(n) \subseteq \mathbb{R}^{n^2}$ is a submanifold...that is, map $A \mapsto A^* A$, and set it equal ...