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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Mass of ellipsoid's surface

Find the mass of ellipsoid's surface $E=\{\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}$ if density $\rho=\frac{r}{4\pi abc}$, where $r=dist(0,T_{(x,y,z)}E)$ and $T_{(x,y,z)}E$ is a surface ...
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1answer
17 views

Can we interpret spherical polars as covering spaces?

My understanding of polar coordinates is that we are implicitly working on a covering space of the punctured plane, given by: $p: \mathbb{R}^+ \times \mathbb{R} \to \mathbb{R}^2 \setminus \{0\} $ $ ...
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1answer
22 views

Line integral segment of parabola

Suppose $$ \vec{F} = \nabla f(x,y) = 6y \sin (xy) \vec{i} + 6x \sin (xy) \vec{j}, $$ and C is the segment of the parabola $y = 5 x^2$ from the point $(2,20)$ to $(6,180)$. Then, what is $$\int_C ...
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2answers
31 views

Volume integral help

I have a volume integral to compute with the following bounded volume $V\in \mathbb{R}^3$ $$ \frac{x^2+y^2}{4}+z^2\leq 1 \;\;,\;\; \frac{1}{2} \sqrt{x^2+y^2}\leq z\;,\;\; z\geq 0$$ I hadn't a clue how ...
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0answers
17 views

Divergence of a triple integral

How to prove the following result (if ture, in the first place)? $$\nabla\cdot\int\int\int ...
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1answer
15 views

Hessian equals zero.

I'm currently just working through some maxima/minima problems, but came across one that was a bit different from the 'standard' ones. So they used the usual procedures and ended up finding that the ...
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1answer
20 views

Conceptual question: Critical Points

This is my first question posted here, I hope to make it as easy-to-answer as possible. I'm currently studying Vector Calculus it is taught that to find critical points (over the entire surface, not ...
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0answers
26 views

Computing the differential (not the Jacobi-Matrix) independent of a basis choice

I am eager to learn more about the differential. I was told that it is a good practice to compute the differential independent of a choice of a basis using the following definition Definition: ...
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2answers
166 views

Green's first identity

Good morning/evening to everybody. I'm interested in proving this proposition from the Green's first identity, which reads that, for any sufficiently differentiable vector field $\mathbf{\Gamma}$ and ...
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1answer
44 views

The differential is NOT the Jacobi Matrix?

In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition: Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to ...
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1answer
37 views

How to evaluate this integral step by step?

Evaluate $$\int\int\sin(x-y)dxdy$$ Is it difficult? see: https://www.wolframalpha.com/input/?i=int+int+sin%28x-y%29+dx+dy
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1answer
25 views

Verify integral over a surface

Show $\int \int_{S} (x^2 +y^2) d\sigma = \frac{9\pi}{4}$ where $S = \left\{(x,y,z) : x>0, y>0, 3>z>0, z^2 =3(x^2+y^2)\right\}$. We have the formula $\int \int_{S} f(x,y,z) = \int \int_{D} ...
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1answer
30 views

Transformation of ellipsoid to sphere

So I need to find an volume-preservating mapping from an ellipsoid to a ball (solid sphere). (Specifically: x^2/9 + y^2 + z^2 <= 3, but I'd rather understand the general case than just get how to ...
3
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1answer
34 views

Separate the variables of the function $\frac{x^2}{\sqrt{x^2+y^2}}$

Is there a way to express the function $\frac{x^2}{\sqrt{x^2+y^2}}$ as the product of two functions: $f(x)\cdot g(y)$, i.e. one in each variable? This is becasue I want to apply a convolution whose ...
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1answer
10 views

Line integral for three line segments

Find where C consists of the three line segments from (2,0,0) to (2,1,0) to (0,1,0) to (0,1,5). I tried to find it like the other line integral but it seems that there is something wrong!
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0answers
14 views

Solution to Tanθ = -3/4 in converting to cylindrical coordinates

I am attempting to convert (8, -6, 7) from rectangular coordinates into cylindrical coordinates. We have r = 10, but then I end up with tanθ = -3/4 and I am not sure how to get an exact answer for ...
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1answer
41 views

A Question about Non-Conservative Vector Fields

In my multivariable calculus class, we spent some time discussing the vector field that was the gradient of arctan(y/x). This field was shown to be non-conservative in closed regions which enclosed ...
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0answers
11 views

Limits and integration

I have the following quick question: Consider bounded open domain $O \subset \mathbb{R}^{n}$ assume that we partition $O$ into $O_{1}^{m}$ and $O_{2}^{m}$ such that $O_{1}^{m},O_{2}^{m} \subset O$, ...
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2answers
27 views

Help understanding question regarding 3rd derivative and “smallest uniform bound”?

I'm a big user of Stack Overflow, however, a first time user here. I'm working on a problem for a math class that's pretty easy (I'm sure), I just don't understand the question really. Here it is ...
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1answer
19 views

Evaluate $I=\int \int _{\delta}(x^my-y^nx)dA$,if $m,n \in \mathbb{N}$ and $\delta$ is the part of the unit disc in the upper half-plane

using polar cordinates i get the following result,by considering a circle $x^2+y^2\le 1$ and $y\ge 0$ $I=\int_{0}^{\pi}\int_{0}^1 (r^m\cos^m\theta r\sin \theta-r^n\sin^n\theta r\sin \theta \cos ...
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2answers
27 views

Can this be written in standard “vector calculus notation”?

A formula for the gradient of the magnitude of a vector field $\mathbf{f}(x, y, z)$ is: $$\nabla \|\mathbf{f}\| = \left(\frac{\mathbf{f}}{\|\mathbf{f}\|} \cdot \frac{\partial \mathbf{f}}{\partial x}, ...
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1answer
39 views

Prove that $\exists x_0 \in D$ such that $f(x_0) = (0,0)$

Here's the question (from last quarter's final): Define $D$ to be the closed unit disk, that is $D = \{(x_1,x_2):x_1^2 + x_2^2 \leq 1\}$. Let $f:D \to \mathbb{R}^2$ be a $C^1$ mapping. If $f'(x)$ ...
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1answer
45 views

Length of a curve in $\mathbb R^n$ smaller than the distance between two points

Let $\gamma : [0,1] \rightarrow \mathbb R^n$ be s.t. $\gamma(0)=a, \gamma(1)=b$ and $\|\gamma' \|\in L^1$. How can I show that $$ \mathscr L (\gamma) = \int _0^1 \| \gamma'(t) \| dt \geq \|a-b\| ...
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2answers
37 views

Need desperate help with sketching functions/equations of functions of 2 and 3 variables

Can someone please give an explanation for the following questions I have just been stuck on this part forever: 1) How do we sketch a function of two variables i.e $f(x,y)=x^2+y^2$ and how do we ...
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1answer
47 views

Line integral with Stokes

Let $C$ be curve $(x-1)^2 + (y-2)^2 =4$ and $z=4$ orientated counterclockwise when viewed from high on the z-axis. Let $$\mathbf{F}(x,y,z)=(z^2 +y^2 +\sin ...
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1answer
23 views

Parametrization of Volume of intersection of two balls

I am trying to find a parametrization of the volume of intersection of two balls $(x-\alpha)^2+y^2+z^2 \leq R^2$ and $(x+\alpha)^2+y^2+z^2 \leq R^2$ , where $R \geq \alpha$, Also , How to find volume ...
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3answers
52 views

$x^2 + y^2 - y = 0$ is… a cylinder?

I've this question: Find the area of the intersection between the sphere $x^2 + y^2 + z^2 = 1$ and the cylinder $x^2 + y^2 - y = 0$. Is this second equation even a closed shape? If one were to ...
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1answer
32 views

help me to find the Line Integral

Find for on the curve counterclockwise around the unit circle C starting at the point (1,0). I dont know the way to do that I tried many ways but I still could not get the right answer
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1answer
7 views

find the line integral

Evaluate the line integral where C is the straight line path from (2,3) to (7, 5). I dont know the way to do that I tried many ways but I still could not get the right answer
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1answer
10 views

Evaluate the Line Integral

Evaluate the line integralwhere and C is given by the vector function , That should be a simple question but I'm getting a wrong answer! The way I did it is evaluating the integral with t instate of ...
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1answer
36 views

Cauchy-Reimann equations and harmonic functions

I have the following question which I am supposed to use the Cauchy-Riemann equations to prove: Let u and v have continuous second derivatives satisfying: $\frac{\delta u}{\delta x} = \frac{\delta ...
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0answers
22 views

Charge Density Triple Integral to infinity

OK, I'm given a charge density function $\displaystyle \frac{2\cdot10^{-4}}{1 + \rho^3}$, and the spherical coordinate of $0 \le \rho < \infty$. Task: find total charge of cloud. Is the answer ...
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2answers
62 views

Newton's Method for Roots of Polynomials

The standard way to use Newton's Method for finding a root of a polynomial $p(x)$ is to use the iteration formula $$x_{n+1}=x_n-{p(x)\over p'(x)}$$ I recently thought of a new way of finding the ...
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2answers
29 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
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1answer
25 views

Wave Equation Descending from 2D to 1D

I am stuck deriving the familiar d'Alembert formula using the Method of Descent, going from 2D to 1D. After using Kirchhoff's Formula, writing the solution independently of $y$ and some manipulations, ...
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2answers
48 views

Solving problems using the *definition* of differentiability

There is a problem in my textbook, that I could not solve and was not able to understand the solution to. The problem had part a, b, c, d. Only a were solved. I am out of luck. I hope, if somebody ...
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1answer
39 views

Double integral and polar coordinates

Please, help me solve this double integral $$\int^{2\pi}_0d\varphi\int^{2}_1\frac{1}{\sqrt{\rho^3\cos^3\varphi+\rho^3\sin^3\varphi}}\rho\,d\rho$$ I really don't know how to figure out and carry of ...
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2answers
39 views

Is it possible to find a function if we know its differential?

Not something we were taught at uni yet, just something that peaked my curiosity. If I was given a derivative of a scalar function, for example $f'(x)=x$ then I know that $f(x)=\frac{x^2}{2}$ (let's ...
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1answer
18 views

double integration via u-subtitution

I'm having trouble with this double integral, maybe someone can help me out: $\int_1^2 \int_0^{lnx} 4x \ dy dx$ My attempt: $$\int_0^{lnx} 4x \ dy = 4xy \big |_{y= 0}^{y= lnx} = 4x \ln(x) $$ $$ ...
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1answer
33 views

Basic surface integral with Stokes.

Calculate surface integral $\iint_S \nabla \times \mathbf{F} \bullet \mathbf{n} \; dS$ with stokes, when $$\mathbf{F}=\left\langle\frac{5y(z-1)}{6},xz, 6e^{xy}\cos{z}\right\rangle$$ and $S$ is surface ...
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0answers
23 views

find the volume of the region using triple integrals using cylindrical coordinates

The volume of the pyramid defined by $(0,0,0)$, $(2,0,0)$, $(0,1,0)$ and $(0,0,4)$. Calculate: $\displaystyle\iiint(2+z^2)\,dV$ The limit of the radius is where I am stumped, since the $xy$ ...
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1answer
19 views

Question with divergence theorem

Calculate flux of field $$\mathbf{F}=(3x^2 y+6)\mathbf{i}+\left(\frac{x y^2 +1}{3}\right)\mathbf{j}+(3yz^2+3)\mathbf{k}$$ in a box where it's opposite angles are in $(1,1,1)$ and $(2,2,2)$ and it's ...
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3answers
74 views

Optimize function on $x^2 + y^2 + z^2 \leq 1$

Optimize $f(x,y,z) = xyz + xy$ on $\mathbb{D} = \{ (x,y,z) \in \mathbb{R^3} : x,y,z \geq 0 \wedge x^2 + y^2 + z^2 \leq 1 \}$. The equation $\nabla f(x,y,z) = (0,0,0)$ yields $x = 0, y = 0, z \geq 0 $ ...
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2answers
21 views

Planes and surfaces and normal vectors?

Is a plane the same thing as a surface? and is the normal vector the same at every point on both??
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0answers
64 views

Understanding double Riemann sums

I have the following two parts of a question, and I merely want to understand what is being asked of me: 1) Divide the region of integration into 4 rectangles of equal width and height. By ...
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1answer
58 views

Integration w/ Change of Variables

folks. I've got this question: Let $D$ be the region $\{(x,y) ~|~ 0 \leq y \leq x, 0 \leq x \leq 1\}$. Evaluate: $$\iint_D (x + y) dxdy$$ by making the change of variables $x = u + v$, $y = u ...
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1answer
32 views

Use Lagrange Multipliers to show the distance from a point to a plane

I'm trying to use Lagrange multipliers to show that the distance from the point (2,0,-1) to the plane $3x-2y+8z-1=0$ is $\frac{3}{\sqrt{77}}$. Our professor gave us two hints: We want to minimize a ...
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1answer
19 views

Evaluate $\iint_s\text{curl}\textbf F\cdot \textbf {n}dS$

Let $\textbf F=<xy,yz,zx>$ and $S$ be the upper half of the ellipsoid $\displaystyle \frac {x^2}{4}+\frac {y^2}{9}+z^2=1$. Evaluate $\iint_s\text {curl}\textbf F\cdot \textbf {n}dS$ I know the ...
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1answer
29 views

Let $S$ be given by $\vec r(u,v)=\langle u\cos v,u\sin v,v\rangle$. Find the tangent plane to the surface at $\vec r(1,\frac {\pi}{4})$.

Let $S$ be given by the vector valued function $\vec r(u,v)=\langle u\cos v,u\sin v,v\rangle$. Find the tangent plane to the surface at $\vec r\left(1,\frac {\pi}{4}\right)$. What I did: $$\vec ...
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2answers
27 views

higher partial derivative

I'm confused here: $$f(x,y) = \sqrt{x^2 + y + 4}$$ I got: $$\frac{\partial f}{\partial x} = x(x^2 + y + 4)^{-\frac{1}{2}}$$ $$\frac{\partial f}{\partial y} = \frac{1}{2}(x^2 + y + ...