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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45 views

surface integral using substitution

I am stuck trying to calculate the following surface integral: $$\int _{R}\int (x+y)^{2}ds$$ over the the following regions: $$0\leqslant x+2y\leqslant 2\: \: \wedge \: \: 0\leqslant x-y\leqslant ...
2
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1answer
14 views

Taking Fourier transform of integral-differential equation

If $u$ is a solution of the equation $$\frac{\partial}{\partial t} u(x,t) + \int_{-\infty}^{\infty} \text{sinc}(x-y) \cdot \frac{\partial^{2}}{\partial y^{2}} u(y,t) \ dy = 0,$$ with initial condition ...
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1answer
27 views

Transform square region to triangular region

How do you express x and y in terms of u and v so that the region $\{(u,v): 0\le u, v\le 1\}$ is mapped to the triangular region in the $xy$-plane with vertices $(0,0)$, $(1,0)$, and $(0,1)$? Now, ...
4
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0answers
19 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. $f$ ...
0
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1answer
27 views

How to calculate the partial derivatives of the composition $F(u(s,t),v(s,t))$?

Could someone help me to understand how to do this problem? I believe Partial Derivatives are used. Thanks!
2
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1answer
20 views

Explanation of Lagrange Equation with Chain Rule

I am just reading through some lecture notes explaining the Lagrange Equation, and I am a bit confused with some chain rule stuff, I get to the part with: $$\frac{\partial F}{\partial y} = ...
0
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2answers
31 views

Shortest distance between a point an a non linear surface

I am trying to find the shortest distance between a point $P(p_1,p_2,p_3)$ and the surface $z=xy$. Could somebody help me please?
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1answer
20 views

Rewrite the following surface so that I can graph it.

$z = \dfrac{1+x^2}{1+y^2}$ $ $ I want the part of the surface above the square $|x|+|y|\leq 1$ $ $ OR we can write this square as $-y<x<y$ and $-1<x<-1$ $ $ I have spent hours trying ...
2
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2answers
27 views

Double integral where limits are the first quadrant

Evaluate the integral $$\iint\limits_D \frac{1}{(x+y+1)^3} \, dA$$ where $D$ is the first quadrant. In this case, what would the limits of integration be? I'm having trouble moving to polar ...
4
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1answer
91 views

Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit ...
0
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0answers
14 views

Volume of a shape using spherical coordinates and integrals

A solid is described in spherical coordinates by the inequality ρ ≤ sin(φ). Find its volume. So, I took ρ from 0 to sin(φ), φ from 0 to pi, and theta from 0 to 2Pi and formed the integral: integral ...
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1answer
26 views

Diagonal matrices and integrals

Suppose that $$A=\int_{\alpha}^{\beta} f(B,x)\ dx,$$ where $B$ is a $3\times3$ matrix. The result I'm looking for is that if $B$ is diagonalized with an orthogonal matrix, then is A diagonalized by ...
2
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2answers
30 views

Is this proof of multivariable limits legit?

Show that the limit $\lim \limits_{(x,y) \to (0,0)} \frac{x^2 - y^2}{x^2 + y^2}$ does not exists. Step 1) let $x=0$ $\Rightarrow \frac{-y^2}{y^2} =1$ Step 2) let $x = y \Rightarrow \frac{0}{2y^2} =0 ...
0
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1answer
32 views

Verification of the Stokes theorem for the surface that is a part of a cone

In the solution to this question in order to work out the surface integral you can project on to $z=3$ and evaluate over the region $x^2+y^2\leq9$. I was just wondering whether or not you could do ...
3
votes
3answers
281 views

Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
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1answer
30 views

Work done by a force field line integrals

Find the work done by the force field $F(x, y) = \langle 2x \sin(y), 2y \rangle$ on a particle that moves along the parabola $y = x^2$ from $(-1, 1)$ to $(2, 4)$. So to use line integrals to solve ...
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1answer
27 views

Minimization with two functions that are not completely related

Two caveats: 1) This is a problem I formulated myself, and so may not be structured correctly/logically. 2) I don't have an extensive math background, but am currently finishing up Calc 3. I have an ...
1
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1answer
58 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
1
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1answer
31 views

Finding a function from a vector field

The vector field $F(x, y) = \left(\displaystyle\frac{x}{r^3}, \frac{y}{r^3}\right)$ appears in electrostatics, where $r = \sqrt{x^2 + y^2}$ is the distance to the charge. Find a function $f(x, y)$ ...
2
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1answer
42 views

Computing double integral

Find $$\iint\limits_D \sqrt{(x-10)^2+y^2}\hspace{1mm}dA$$ where $\{(x, y)\in D \mid x^2+y^2\leq 10^2\}$. I am not sure how to start, every way I have tried so far, ends up into something ugly. All ...
3
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2answers
233 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
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2answers
58 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
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2answers
22 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
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2answers
435 views

Prove $f(x,y) = xy/(x^2 + y^2)$ is continuous everywhere except $(0,0).$

I'd just like to ask you if my proof here is valid. I'll provide you with the method I used and if it seems ok let me know! If not, explanations would be helpful! My main approach to this question ...
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0answers
8 views

trace of hyperbola 2 sheets

for yz parallel trace book mentions there is no trace unless |x|>|a| then the trace is an ellipse it is hard for me to visualize this any insight would be appreciated, why is this so? sorry I know ...
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2answers
60 views

Convergence of $\int (1+|x|)^{-\alpha} dx$

I want to prove that $$\int_{\mathbb R^n} (1+|x|)^{-\alpha} dx < \infty$$ if and only if $\alpha > n$, but I have no idea how to generally prove this. It is easy to see for $n=1, 2 \ldots$ but I ...
1
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1answer
37 views

Bounding the integral of a function by the integral of its derivative

I have no idea where to begin for this question, so any help would be greatly appreciated! Let $\Omega$ be a square with side 1. Show that $$\left(\int_{\Omega} v^2 \, dx \right)^{1/2} \leq \left( ...
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0answers
52 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
3
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1answer
37 views

Sketching a surface

I'm trying to draw a sketch to get a feel of the situation but am confused as to what the question is asking. I have sketched $y^2=8x$ in the plane $z=0$ and marked on the points where $y$ and $z$ ...
1
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1answer
18 views

Derivation of Mahalanobis Distance

I was recently reading up on the Mahalanobis Distance, and understood how it generalizes distance measures for multivariate data such as the Euclidean Distance. However, what got me wondering was how ...
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3answers
81 views

How do I find a point on the surface of a sphere

How do I find a point on a sphere knowing its radius and center point ? I have a sphere: $$x^2+(y-1)^2+(z+3)^2=16$$ Obviously its center point is $(0,1,-3)$ and its radius is $4$. I am asked to find ...
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1answer
30 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
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1answer
40 views

Use triple integrals to integrate over a tetrahedron

Integrate $f(x, y, z) = x^2 + y^2 - z$ over the tetrahedron with vertices $(0, 0, 0), (1, 1, 0), (0, 1, 0), (0, 0, 3)$. I need to use triple integrals to solve this, so I made a diagram and set the ...
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2answers
29 views

Parametrization Question

When computing a line integral, or any integral that requires parametrization, what are you integrating with respect to? For example, if parametrizing in polar coordinates, with $x=r\cos\theta$ and ...
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1answer
17 views

Average value for multiple integrals

If there is a function $f(x,y)$ and we want to find the average value over a region $R$ defined by $0<x<1$ and $0<y<x$, how is that computed? I know that it would be something like this: ...
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1answer
12 views

Reference for transformation of integrals over Lipschitz boundaries

Let $D\subseteq\mathbb{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Then according to page 314 of [Function spaces, Alois Kufner, 1977] one can define a surface integral of a real valued function ...
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2answers
34 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
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1answer
264 views

taylor's formula in $\mathbb{R}^n$ as exponential of derivative operator

I hope this question is not too vague. There is some relationship between the formal operator $$e^\frac{d}{dx} = 1 + \frac{d}{dx} + \frac{1}{2!}\frac{d^2}{dx^2} + \ldots $$ and taylor's formula. ...
1
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1answer
66 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
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1answer
31 views

Parametrizing to Calculate Flux

Evaluate the flux of $\mathbf{f}$ across the oriented surface $\Sigma$ by computing the surface integral $\iint_{\Sigma} \mathbf{f} \cdot d\sigma$, where $\Sigma$ is the surface $z=xe^y$ for $0 \leq x ...
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0answers
82 views
+50

quasi-convexity of a function

Can someone help me identify whether the following function is quasi-convex? Let $p>1$. For $x=(x_1,\dots,x_n),x_i>0,\sum_ix_i=1$, we define $$f(x) = -\log \sum_i (x_i/\|x\|_p)^{p-1}.$$ Plots ...
1
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1answer
22 views

Tangent line to a curve statement

I am having problems understanding some parts of the proof of some statement related to tangent line to a curve. I'll copy the exact statement and proof and then my doubts. Statement If $\mathcal C$ ...
3
votes
2answers
77 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
3
votes
2answers
36 views

how to prove gradients vectors are the same in polar and cartesian coordinates.

Suppose $T=T(r,\theta)=G(x,y)$ How do you prove $\nabla T(r,\theta)=\nabla G(x,y)$? I can think of some arguments in favor of this equality, but I want an actual proof or a very good intuitive ...
3
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0answers
30 views

Does chain rule require continuously differentiability?

Recently I read the book Advanced Calculus written by Fitzpatrick. The Theorem 15.34 tells that If $F:\mathbb R^n\to\mathbb R^m$ is CONTINUOUSLY differentiable (all partial derivatives exist and ...
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2answers
33 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
2
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1answer
25 views

Transformation rule for partial derivatives

I can't fathom the step I have highlighted in green. Am I using the chain rule in 3 dimensions? What is it that I am transforming here?
2
votes
2answers
23 views

Confusion about Spherical Coordinates Transformation

We have a function $$f(x,y,z) = \frac{e^{-x^2 -y^2 -z^2}}{\sqrt{x^2+y^2+z^2}}$$ and we want to integrate it over the whole $\mathbb{R}^3$. Then what i got is the following: $$\int_{\mathbb{R}^3}^ \! ...
0
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1answer
26 views

Directional Derivative Derivation

I don't understand the part underlined in the derivation of the directional derivative. Why is the $\lim_{Q \to P}$ interchangeable with $\lim_{N \to P}$? I understand that the surfaces are getting ...
4
votes
1answer
403 views

local diffeomorphism

I hope this finds you all well. Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in ...