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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2
votes
1answer
415 views

Express partial derivatives of second order (and the Laplacian) in polar coordinates

$z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$ Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$ I'm having big troubles with using chain rule, in particularly the second ...
2
votes
2answers
245 views

Question about Implicit function theorem

I was asked a simple question, show that $y+\sin y=x$ sets in the neighborhood of $(0,0)$ $y$ as a function of $x$, and find $\dfrac{dy}{dx}(0,0)$ Firstly, my naive solution would be: Since $lim_{y ...
2
votes
4answers
7k views

Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$

"Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$" $$P = (1,2,1), Q = (0,4,0), A = (8,3,2)$$ $OP$ = vector to $P$ $$PQ_ = (0,4,0) - ...
2
votes
4answers
450 views

Find the limit (if it exists): $\lim_{(x,y,x)\rightarrow(0,0,0)}\frac{xyz}{x^2+y^2+z^2}$

Find the limit (if it even exists). If not, prove it doesn't exist. $$\lim_{(x,y,x)\rightarrow(0,0,0)}\frac{xyz}{x^2+y^2+z^2}$$
2
votes
1answer
686 views

Computing $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$

What is the result of $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$ . I tried to do couple of algebraic manipulations, but I didn't reach to any conclusion. Thanks a lot.
1
vote
2answers
122 views

Proof Strategy - Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = yz^2\mathbf{i}$

2013 10C. Question: Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable ...
1
vote
2answers
137 views

$\theta$ for Triple Integral above paraboloid $z = x^2 + y^2$ and below $z = 2y$ [Stewart P1011 15.8.37]

$\bf\sf37.$ Evaluate $\iiint_E z\,dV,$ where $E$ lies above the paraboloid $z=x^2+y^2$ and below the plane $z=2y.$ In cylindrical coordinates the paraboloid is given by $z=r^2$ and the plane by ...
1
vote
3answers
306 views

Show that $g(x,y)=\frac{x^2+y^2}{x+y}$ is continuous at $(0,0)$.

Let $g:\mathbb{R^2}\rightarrow \mathbb{R} $ so that, in $M=[0,1]\times[0,1]$, $$g(x,y)=\begin{cases}\frac{x^2+y^2}{x+y} &\text{ if }x+y \neq 0,\\\\ 0&\text{ if }x+y=0\end{cases}$$ Show that ...
1
vote
2answers
374 views

Do we need additional assumptions for problem 3-37 (b) in Spivak´s calculus on Manifolds?

Problem 3-37 (b) reads: Let $A_{n}=[1-1/2^{n},1-1/2^{n+1}]$. Suppose that $f:(0,1)\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for any $x\notin$ any $A_{n}$. Show that ...
1
vote
0answers
185 views

Does this hold?

Strayed on the following question. Assume that $x_{1}$,$\ldots$, $x_{d}\ge0$ with $x_{1}+\ldots+x_{d}=1$ and $y_{1},\ldots,y_{d}\in\mathbb{R}$. Does $$ \min_{1\le i\ne j\le ...
0
votes
3answers
111 views

How to differentiate the following interesting vector product?

How do we differentiate the following vector product with respect to $\boldsymbol r$. \begin{equation} \frac{d}{d\boldsymbol r}\bigg[(\boldsymbol \omega \times\boldsymbol r)\cdot (\boldsymbol \omega ...
0
votes
3answers
4k views

How to prove this vector identity [closed]

How do i prove this vector identity ? $$(\vec a \times \vec b)\times \vec c=(\vec a \cdot\vec c)\vec b - (\vec b\cdot\vec c)\vec a$$
0
votes
2answers
131 views

How does partial derivative work?

I don't understand the second step at all. Where did the $\partial^2 u/ \partial x^2$ come from and why do we have six terms?
12
votes
5answers
40k views

Surface Element in Spherical Coordinates

In spherical polars, $$x=r\cos(\phi)\sin(\theta)$$ $$y=r\sin(\phi)\sin(\theta)$$ $$z=r\cos(\theta)$$ I want to work out an integral over the surface of a sphere - ie $r$ constant. I'm able to derive ...
16
votes
1answer
2k views

Nice way of thinking about the Laplace operator… but what's the proof?

Here's a nice fact: roughly speaking, the Laplace operator gives you the difference between the value of a function at a point and the average value at "neighboring" points. More precisely, in ...
13
votes
1answer
826 views

Visualizing Commutator of Two Vector Fields

I'm reading a book on calculus, the part about vector fields on manifolds. It's a nice book, but with a severe drawback --- it has no pictures. I like how vectors are treated algebraically, as ...
10
votes
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! ...
9
votes
1answer
175 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) ...
8
votes
1answer
1k views

If derivative of a function is the zero function in $\mathbb R^n$, then the function is constant when the domain is path-connected

Some definitions first. Let $A \subseteq \mathbb R^n$. Let $x,y \in A$. A path between $x$ and $y$ is a continuous function $f: [0,1] \rightarrow \mathbb{R}^n$ with $f(0) = x$ and $f(1) = y$. The set ...
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 ...
22
votes
5answers
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. ...
6
votes
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 ...
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 ...
11
votes
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. ...
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
667 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
395 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
457 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 $\lambda_1, \lambda_2, ..., \lambda_n$ ...
3
votes
2answers
699 views

A curve parametrized by arc length

Let $C$ be a plane curve parametrized by arc length by $\alpha(s)$, $T(s)$ (unit tangent vector) and $N(s)$ (unit normal vector). Prove that $$\frac{d}{ds} N(s)=-\kappa(s)T(s).$$ I know that ...
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 ...
13
votes
4answers
18k views

L'hospital rule for two variable.

How to use L'hospital rule to compute the limit of the given function $$\lim_{(x,y)\to (0,0)} \frac{x^{2}+y^{2}}{x+y}?$$
11
votes
1answer
735 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 ...
11
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 ...
9
votes
2answers
939 views

Partition of Unity in Spivak's Calculus on Manifolds

I have a question about partitions of unity specifically in the book Calculus on Manifolds by Spivak. In case 1 for the proof of existence of partition of unity, why is there a need for the function ...
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
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 ...
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) + ...
7
votes
1answer
161 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
73 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 ...
6
votes
2answers
5k views

Vector sum in spherical coordinates

I can't seem to come up with a simple formula to head-tail adding two vectors in spherical coordinates. So I'd like to know: Can anybody point out a way to do it in spherical coordinates (without ...
5
votes
1answer
777 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 ...
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
565 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 ...
4
votes
2answers
578 views

Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that $u \in C^2(\overline{B}_r)$ satises \begin{align} -\Delta u(x)=f(x), & \qquad ...
4
votes
1answer
241 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 ...
1
vote
3answers
119 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
11
votes
3answers
390 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 ...
9
votes
1answer
5k views

Solving quadratic vector equation

Hope it is a right place to ask how to solve the equation on $\mathbf x$: $$ \mathbf x^T \mathbf A\mathbf x + \mathbf x^T \mathbf b + c = 0. $$ where: $\mathbf x$ is an $n\times 1$ column vector ...