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
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1answer
52 views

Double Integral Confusion

A buddy was asking me for help with one of his MV Calc problems, and I ended up getting the same answer as him so I figured I'd ask it here... Question Find $$\iint_{R} (x-1) \, dA$$ where $R$ is ...
0
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2answers
230 views

Let $R$ be the square with vertices $(0,0), (1,-1), (2,0), (1,1)$. Evaluate $\iint_R\sqrt {\frac{x-y}{x+y+1}}$.

Let $R$ be the square with vertices $(0,0), (1,-1), (2,0), (1,1)$. Evaluate $\displaystyle \iint_R\sqrt {\frac{x-y}{x+y+1}}$. I realized one way to approach this problem is to divide it into 4 equal ...
1
vote
1answer
40 views

Let $f(x,y)=\frac {x^2}{2}+\frac {y^2}{4}$ on $\{(x,y)|x^2-y^2=2\}$. Find the absolute maximum and minimum if they exist.

Let $f(x,y)=\frac {x^2}{2}+\frac {y^2}{4}$ on $\{(x,y)|x^2-y^2=2\}$. Find the absolute maximum and minimum if they exist. I approached this problem using lagrange multiplier, with $g(x,y)=x^2-y^2-2$. ...
5
votes
1answer
403 views

Finding the volume inside an elyptical cylinder and a sphere

I'm trying to find the volume bounded by a sphere and an elyptical cylinder. The sphere is given by $x^2+y^2+z^2=1$ and the elyptical cylinder by $2x^2+y^2-2x=0$. My first attempt with spherical ...
11
votes
1answer
137 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
4
votes
3answers
314 views

Multivariable limit with polar coordinates

Polar coordinates do not reveal the behaviour of $f(x,y)$ when studying $$ \lim_{x^2 + y^2 \to \infty} \frac {xy}{e^{x^2y^2}} $$ In polar coordinates we have $$ \lim_{r^2 \to \infty} \frac 12 \frac ...
0
votes
1answer
100 views

check if complex function is differentiable

The question is to check where the following complex function is differentiable. $$w=z \left| z\right|$$ $$w=\sqrt{x^2+y^2} (x+i y)$$ $$u = x\sqrt{x^2+y^2}$$ $$v = y\sqrt{x^2+y^2}$$ Using the ...
2
votes
1answer
451 views

A silly mistake concerning spherical coordinates and unit vectors…

I'm quite comfortable with vector calculus in all sorts of coordinate systems, but for the love of me, I can't seem to figure out where did I go wrong in this simple derivation of the position vector ...
2
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0answers
42 views

Cancelling some dx's

How can you prove $$\frac{\partial f}{\partial y} = \frac{d}{d x} \left( \left( \frac{\partial}{\partial \frac{dy}{dx} } \right) \right)f ?$$ It's tempting to cancel the two $dx$'s, but can ...
0
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2answers
68 views

Multivariable Calculus Volume

I have to find the volume inside a cylinder with radius 1 ($x^2+y^2=1$) which is bounded by the planes $z=4-2x-y $and x=0, y=0, z=0. I am not sure what to do but I suspsect it requires converting to ...
2
votes
0answers
66 views

Area between two curves with a certain domain.

I am trying to find the area between two curves over a certain domain. The region of integration is between $xy=5$, $x=9-y^2$ and the lines $y=1$ and $y=2$. I have to show that this can be written as ...
0
votes
1answer
70 views

Evaluate $\iint_D\sin(xy)dA$ where $D$ is bounded by $y=\frac 1x, y=\frac2x, y=x, y=2x$ in the first quadrant.

Evaluate $\iint_D\sin(xy)dA$ where $D$ is bounded by $y=\frac 1x, y=\frac2x, y=x, y=2x$ in the first quadrant. By subbing numbers into the equation, I see that $1\leq x\leq 2, 1\leq y\leq 2.$ ...
3
votes
1answer
54 views

Studying the differential form $w(x,y)=xy^ndx+x^mydy$

Let $w$ be a differential form defined by $w(x,y)=xy^ndx+x^mydy$ where $m$ and $n$ are non negative integers. 1) For which values of $m$ and $n$ the differential form $w$ is closed? 2) For these ...
1
vote
1answer
44 views

Show, by finding a potential V(r) such that F = −∇V , that F is conservative

A particle at position r experiences a force: $$F=(-\frac{a}{r^2}+\frac{b}{r^3})\hat{r}$$ a and b are constants and $\hat{r}$ is the unit vector in a radial direction. I am told that I will need the ...
2
votes
1answer
116 views

Triple integral with a cone as a domain

How can I find $\displaystyle\iiint_D f$ if $f(x,y,z) =\sqrt{x^2+y^2}$ $D$ is what is inside of $z^2=x^2+y^2,z=0,z=1$?. I tried to do it with cylindrical coordinates as follows: $x=\rho\cos\theta, ...
1
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0answers
44 views

Help to understand how the chain rule works

Suppose we have a function $U_d(q_1-q_{j1},q_2-q_{j2})$ where $q_1,q_{j1},q_2,q_{j2} \in \mathbb{R}$ and $q_1=q_{d1}+q_{j1}, q_2=q_{d2}+q_{j2}$. Why is $\frac{\partial U_d}{\partial ...
0
votes
1answer
108 views

Is it right to try finding a vector where the dot product is zero in this problem?

In an assignment, I am given this problem: A blimp is held in place with two ropes. The wind is creating a force of 300 N on a bearing of 280° and one of the ropes is exerting a force of 250 N on ...
1
vote
1answer
55 views

Integration with cylindrical coordinates: Should I split the integral?

I have $f(x,y,z) = x^2+y^2$ and $D=\{(x,y,z): (x,y,z) \text{ are points inside } x^2+y^2=2x \text{ and between} z=0,z=2\}$ The equation $x^2+y^2=2x$ is equivalent to $(x-1)^2+y^2=1$ which ...
4
votes
2answers
106 views

Show that if a function $f : \mathbb{R}^n \to \mathbb{R}^m$ is differentiable with differentiable inverse then $m = n$

So far I have: $\boldsymbol{f^{-1}} \circ \boldsymbol{f}(\boldsymbol{a}) = \boldsymbol{a} \implies [\boldsymbol{D}(\boldsymbol{f^{-1}}(\boldsymbol{a}) \circ \boldsymbol{f}(\boldsymbol{a}))] = I_n ...
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0answers
79 views

An attempt to a triple integral with spherical coordinates.

If $f(x,y,z)=xyz$ and $D=\{(x,y,z)\in\mathbb{R}^3:x\geq 0, y \geq 0, z \geq 0,\; x^2+y^2+z^2\leq 1\}$, find $$ \iiint_D f(x,y,z). $$ I tried to solve this with spherical coordinates: ...
3
votes
2answers
195 views

Find the volume between two surfaces

Find the volume between $z=x^2$ and $z=4-x^2-y^2$ I made the plot and it looks like this: It seems that the projection over the $xy$-plane is an ellipse, because if $z=x^2$ and $z=4-x^2-y^2$ ...
1
vote
1answer
660 views

How is the entropy of the multivariate normal distribution with mean 0 calculated?

Here is what I have so far: $$\begin{align} h(x) &= - \int \frac{1}{(2\pi)^{\frac{D}{2}}\det\Sigma^{\frac{1}{2}}} \exp(-\frac{1}{2} x^T\Sigma^{-1}x) \ln ...
0
votes
1answer
26 views

gradients 2 variable

Find $\phi$ , so that $F(x,y)=\nabla \phi (x,y)$ When $F(x,y)=(y\cos x, \sin x +\frac{2y}{3} e^{y^2})$ and when $F(x,y)=(\frac{1}{xy}+\frac{2x}{1+x^2 y}, \frac{1}{xy}+\frac{x^2}{1+x^2 y})$
2
votes
1answer
59 views

Proving Borsuk-Ulam with Stokes

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem? So I want to prove the following: Given a map $f\colon S^2\rightarrow ...
9
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1answer
1k views

Gradient Descent on Non-Convex Function Works But How?

For Netflix Prize competition on recommendations one method used a stochastic gradient descent, popularized by Simon Funk who used it to solve an SVD approximately. The math is better explained here ...
1
vote
1answer
160 views

Finding the Boundary Conditions for a Laplace's Equation in Polar Coordinates

I have solved Laplace's equation in Polar Coordinates for the scalar electric potential in a circle of radius R and have the solution $$ \phi(r,\varphi) = \phi_{0} + ...
3
votes
0answers
850 views

Stuck on derivation of divergence in cylindrical coordinates

I'm having a hard time trying to derive the divergence in cylindrical coordinates from its expression in cartesian coordinates $\frac {\partial F_{x}} {\partial x}+\frac {\partial F_{y}} {\partial ...
3
votes
1answer
182 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} ...
2
votes
1answer
78 views

Finding a volume

Find the volume of $D\{(x,y,z)\in \mathbb{R}^3:\frac{x^2}{a^2} +\frac{y^2}{b^2}\leq z\leq 1 \}$ It looks like (1) I believe this could be solve with a double integral an considering the ...
1
vote
1answer
36 views

Potential in $2$ dimensional systems

Given a $1$ dimensional dynamical system represented by $\dot{x}=f(x)$ we define the potential $V(x)$ to be the function that satisfies $\dot{x}=f(x)= -\frac{\partial V(x)}{\partial x}$. How to we ...
0
votes
3answers
4k views

Finding volumes - when to use double integrals and triple integrals?

This is not a technical question at all, but I'm quite confused about what should I use to compute volumes in $\mathbb{R}^3$ with integration. I've read somewhere that a double integral gets the ...
-2
votes
1answer
58 views

Need a translation: Mathspeak to english

Watching a video on multiple integration. Maybe its that the coffee has not kicked in but I am having trouble with understanding the graphing term "mapping". Can anyone put it in layman's terms or at ...
0
votes
1answer
37 views

Question about limit.

My question is that in my practical sheet I have been given a question which says show that limit doesn't exist and question is $f(x,y)= \frac{x^2}{x^2+y^2-x}$ s.t $(x,y)\ne(0,0)$ My question is: ...
0
votes
1answer
63 views

Differentiation with help of Frenet Frame

Show that if $(\frac{1}{k})^{'} \neq 0$ and $(\frac{1}{k})^2 + ((\frac{1}{k})^{'}\frac{1}{\tau})^2$ is a constant, then a unit speed curve $\alpha$ lies on a sphere. Using the following formulas ...
3
votes
1answer
117 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
1
vote
1answer
23 views

differential split of $2$ variable problem

$d(x_1x_2)= x_1dx_2 + x_2dx_1$ as given in theory now $\int d(x_1x_2)= \int x_1dx_2 + x_2dx_1$ but integrating both the sides give $x_1x_2 = 2x_1x_2 ..$ why ? I guess I am missing something very ...
0
votes
1answer
31 views

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ [duplicate]

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ $\displaystyle \int_1^2\int_0^1\frac ...
2
votes
3answers
89 views

Why is the value of this line integral constant

Consider the line integral given by $$\int_C \frac{(x+y)\,dx-(x-y)\,dy}{x^2+y^2}$$ where $C$ is any simple closed curve around the origin. Can someone explain, without using complex analysis, why this ...
1
vote
1answer
91 views

How is statistical uncertainty calculated for the modulus function?

I know it's an unusual function to calculate an uncertainty for, but I haven't been able to figure out a reasonable means for calculating derivatives for it to do so myself. I know modular arithmetic ...
1
vote
1answer
29 views

Proof for $\int_{S^n(r)} \, dx=\int_{S^n(1)}r^n \, dx$?

I read the wikipedia article about $n$-sphere. I'm trying to give a proof for the following formula $$ \int_{S^n(r)}dx=\int_{S^n(1)}r^ndx,\tag{*} $$ where $S^n(r):=\{x\in {\Bbb R}^{n+1}:|x|=r\}$ for ...
2
votes
2answers
1k views

Multidimensional Fourier transform of the laplacian

In my course on electromagnetic field theory we use the Fourier transform to simplify Maxwell's equations, for example: $$\frac{\partial ^2\vec E(\vec r,t)}{\partial t^2} \rightleftharpoons ...
3
votes
2answers
172 views

Double integral — tricky?

If $f(x,y) = x^2+y^2$ and $D=\{(x,y)\in\mathbb{R}^2:x^2+y^2\geq1, x^2+y^2-2x\leq0 \text{ and } y\geq0\}$, find $\displaystyle\int\displaystyle\int_D f$. $D$ looks like the intersection between ...
0
votes
1answer
65 views

Simple integration of a differential along 3 separate paths

So I have $$dw = \frac{y}{a}dx + \frac{x}{a}dy$$ and points $$A=(0,0), B=(1,0), C=(1,1), D=(0,1)$$ How do I integrate along paths ABC and ADC? and how can I change variables integrate along the ...
3
votes
1answer
185 views

Find the critical points for $F(x,y,z)=-x^{3}-y^{2}+2xy+x+2z$

I started by taking the first order partial derivatives: $F_{x}=-3x^{2}+2y + 1$ $F_{y}=-2y+2x $ $F_{z}=2 $ Now I would try to solve it for $F_{x}=F_{y}=F_{z}=0$ but $F_{z}=2$. How can I proceed or ...
0
votes
1answer
88 views

Mapping Confusion -Implicit Function Theorem-

Here is the Implicit Function Theorem statement: "Let $g : R^k \times R^n \to R^n$ be a continously differentiable function s.t. $g(x_0, y_0) = c$ and $D_yg(x_0,y_0) : R^n \to R^n$ is an isomorphism. ...
3
votes
2answers
253 views

Solution of a partial differential equation.

Find $u$ if $$\dfrac{\partial^2 u}{\partial x^2} = 6xy, \,\,u(0,y) = y, \,\,\dfrac{\partial u}{\partial x}(1,y)=0.$$ I have tried by laplace transformation $$\displaystyle ...
0
votes
3answers
218 views

Derivatives of multivariable functions

I would like to make a few statements about a simple object - the derivative of a univariate function - and apply and relate its features and my understaning of them to multivariate functions. ...
1
vote
1answer
96 views

Arclength does not change with reparametrization

Recall that the length of a curve $\alpha : [a,b] \rightarrow \mathbb{R}^3$ is given by $L(\alpha) = \int |\alpha'(t)| dt$. Let $\beta(r): [c,d] \rightarrow \mathbb{R}^3$ be a reparametrization of ...
4
votes
0answers
140 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
0
votes
3answers
55 views

A question about a simple integral.

How could I show that the $$\iint\sin(x)dxdy$$ along the domain $$x^2+y^2\leq1$$ is zero? I tried using polar coordinates but to no avail. Had thought about claiming Sine is an odd function so the ...