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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0
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1answer
415 views

Compatibility Condition of the Poisson Equation with Neumann Boundary Conditions

I am trying to solve the following general Poisson equation with homogeneous Neumann boundary conditions in a rectangular domain ($0 \le x \le L$ and $0 \le y \le H$). $$ \frac{\partial^2 ...
1
vote
1answer
30 views

How to apply the Implicit function Theorem in multi-variable Calculus

First of all, I'm sorry for any English mistakes I might make, it's not my first language. I have some trouble understanding how to apply the implicit function theorem when there are several equations ...
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1answer
17 views

Is a function which is bounded along a path constant? [on hold]

Does boundedness along a path of a multivariable function imply that the function is bounded?
-1
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0answers
11 views

Tricky Multivariable Integral

Evaluate the integral $\int \cdots\int_{x_1+x_2+\ldots+x_n\le M \text{and} x_1,\ldots,x_n\ge0}\frac{1}{x_1+x_2+\ldots+x_n}dx_1\ldots dx_n.$
2
votes
1answer
24 views

$A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
0
votes
1answer
26 views

Solving the integration problem by use of fundamental theorem of calculus and chain rule

In one test the question said $$f(x,t)=\int_{0}^{g(x,t)} e^{-u^2} du$$ Now how I can calculate $\partial^2f/\partial t^2$ ? I have this idea: $$\int_{0}^{g(x,t)} e^{-u^2} du=F(g(x,t))-F(0)$$ ...
1
vote
1answer
39 views

Maximum of function containing two variables $x$ and $y$

If $x+y+\sqrt{2x^2+2xy+3y^2} = k(\bf{Const.})\;,$ Then $\max(x^2y)\;,$ Where $x,y\geq 0$ $\bf{My\; Try::}$ Let $x^2y=z\;$ Then we get $$x+\frac{x^2}{z}+\sqrt{2x^2+\frac{2z}{x}+\frac{3z^2}{x^4}} = ...
1
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1answer
8 views

Multivariate Non-Differentiability

This example says that "continuous partial derivatives imply differentiability but not vice-versa". Based on transposition logic, I would then assume that if a multivariate function has discontinuous ...
0
votes
2answers
43 views

A problem on norm preserving and angle preserving and their relations.

I want to solve the following problem and finding some difficulties:- I have done the part (a) easily. My problem is in part (b) and (c). In part (b) after calculation I have achieved that ...
0
votes
1answer
17 views

If the integral over an area is zero is the integral of the gradient also zero?

Say I know that $\int\int v_z dx dy = 0$ over some area with $dA = dx dy$. $v_z$ is a function of $x$ that "points" in $z$. Is this enough to say that $\int\int \frac{\partial v_z}{\partial x} dx dy = ...
0
votes
0answers
9 views

Is this curl operator with surface normal and tangential components valid?

Is this curl operator valid? $\nabla \times \mathbf{A} = (\partial_{\tau_1} A_{\tau_2} - \partial_{\tau_2} A_{\tau_1}) \hat{\mathbf{n}} - (\partial_n A_{\tau_2} - \partial_{\tau_2} A_n) ...
-1
votes
1answer
24 views

Calculate the Area of Toroidal

i need help with the integral of this function. $ f(x,y)=\left( x-b \right) ^{2}+{y}^{2}={a}^{2}$ Note: $0 < a < b$ I know the graphics of this function is this ...
0
votes
1answer
14 views

Calculating interesting volume (Also general multiple variable function equation)

First off, What does $f(x,y,z) = 1$ mean? versus $f(x,y,z) = x$. I have trouble with this as I am not sure choosing an initial $x$ factors into the function. Similarly with choosing a corresponding ...
0
votes
0answers
12 views

Solenoidal vector field

I am to prove (using the equations for gradient, divergence and curl in spherical polar coordinates) that vector field $\mathbf{w}=w_{\psi}(r,\theta)\hat e_{\psi}$ is solenoidal, find ...
2
votes
0answers
67 views

If $\int_{y=0}^{y=b}\int_{x=0}^{x=a}f(x,y)dxdy=ab$, what is $f(x,y)$?

If $\int_{y=0}^{y=b}\int_{x=0}^{x=a}f(x,y)dxdy=ab$, for all $a,b$ between $0$ and $1$, what is $f(x,y)$? I know that the answer is 1. But I want to know how you would get there, if you didn't know ...
0
votes
0answers
7 views

Square-integrability in one variable of two-variables function

Let $h: [0,T]^2 \to \mathbb{R}_+$ be a measurable function. Assume that $$ \int_0^T\left(\int_0^T h(r,s)ds\right)^2dr < \infty .$$ Does this imply that $$ \int_0^T \left(h(r,s)\right)^2 dr < ...
17
votes
13answers
3k views

Why does a distance and its square reach their minimum at the same point?

There is a question in my calculus textbook that asks to find a point on the parabola $y^2 = 2x$ that is closest to point $(1,4)$. They want us to first use the distance formula, but then proceeded ...
0
votes
3answers
87 views

Finding the Shortest Distance from Point to Plane

I am trying to find the shortest distance from the point (3,0,-8) to the plane x+y+z = 8 and I keep getting the same incorrect solution. First, I found the equation fo the distance to be: ...
0
votes
1answer
43 views

How to solve equations of the form $ax^4+bx^2+c$

Question: Find the values of $x$ for the following equation $$108x^4-507x^2+300$$ My attempt: I have the solution and it says to use the quadratic formula. However how would I apply that here? ...
0
votes
0answers
17 views

Parametrising a side of a cuboid

Question: Suppose the surface S is bounded by 6 planes $$x=0,x=2,y=0,y=4,z=0,z=1$$ Parametrise two of the surfaces. My attempt: S0 I picked the "floor" face of the cuboid i.e. $x= 0, x=2, y=0,y=4, ...
0
votes
0answers
16 views

Double Integral with Residues

I'm trying to solve the integral $$\int_a^b\int_a^b\frac{dxdy}{1+\left(x^2+y^2\right)^\alpha}$$ where the constant $\alpha$ is real-valued and in the range $\alpha\in[1/2,\infty)$. The bounds $a$ ...
3
votes
2answers
29 views

Implicit derivation to find $\partial x/\partial v$?

I saw this question: $$\begin{cases} x^2+y^2=u \\ x\sin y+y=v\end{cases}$$ What is the $\partial x/\partial v$? I think it should be $1/\sin(y)$ because $\partial v/\partial x=\sin y$, but the ...
0
votes
1answer
23 views

Tangent plane of a surface at points with given gradient

So I'm stuck on the next problem: I need to find the tangent plane of the surface $$u=\ln\left( x+\frac{1}{y} \right)$$ at all the points where the gradient is equal to $$\nabla u=\hat ...
0
votes
1answer
408 views

Triple integral - wedge shaped solid

Find the volume of of the wedge shaped solid that lies above the xy plane, below the $z=x$ plane and within the cylinder $x^2+y^2 = 4$. I'm having serious trouble picturing this. I think the z ...
1
vote
1answer
24 views

Unit base vectors in a new coordinate system

Let's assume we have a function $f:\Omega =R^2 \rightarrow R $ $f(x,y)=x+2xy+x^2y$. Obviously our unit base vectors on $\Omega$ are $e_x=\hat{i}$ and $e_y=\hat{j}$. Now we want to change the ...
2
votes
0answers
76 views

Tricky proof of a result of Michael Nielsen's book “Neural Networks and Deep Learning”.

In his free online book, "Neural Networks and Deep Learning", Michael Nielsen proposes to prove the next result: If $C$ is a cost function which depends on $v_{1}, v_{2}, ..., v_{n}$, he states that ...
2
votes
1answer
181 views

Gradient descent with adaptive learning ratio.

I have a neural network, trained with SGD (stochastic gradient descent) with learning ratio $\alpha$. Each iteration I try to recalculate the weights with a rule: $$\Delta \vec{w} = -\alpha ...
2
votes
0answers
16 views

Multiple vs single change of variable (in convolution)

Let $f$ and $g$ be in $L^1(\mathbb{R})$. Then $$\int_{\mathbb{R}^2}|f(y)g(x-y)|\, dy\, dx=\int_{\mathbb{R}^2}|f(y)| |g(t)|\, dy\, dt=\|f\|_1\|g\|_1$$ This calculation appears in proving that $f*g\in ...
0
votes
0answers
18 views

Prove that integration of a differential $k$-form is independent of choice of basis

This is Exercise 4 of Section 33 of Munkres' "Analysis on Manifolds" book: (Let $A$ be an open set in $\mathbb{R}^k$.) If $\eta$ is a $k$-form in $\mathbb{R}^k$ and if ...
1
vote
2answers
895 views

Hessian after coordinate changing

Let $f\colon \Bbb R^n\to\Bbb R$. Let $z=Px$ coordinate changing. $P$ is $n\times n$ constant matrix, $x$ and $z$ are the variables in $\Bbb R^n$. Does anyone know a formula which express how the ...
0
votes
1answer
22 views

Finding the volume of the region bounded by $z=\sqrt{\frac{x^2}{4}+y^2}$and $x+4z=a$. Cilindrical coordinates.

I have to find the volume of the region bounded by $z=\sqrt{\frac{x^2}{4}+y^2}$and $x+4z=a$. So, here we have a cone and a plane "cutting" it. I definitely must do this using some some of coordinate ...
0
votes
1answer
30 views

Question the convergence of $\iiint_{x^2+y^2+z^2\geq1}\frac{e^{\sin(x+y+z)}}{(x^2+y^2+z^2)^p}$ in dependence of $p$

Question the convergence of $$\iiint_{x^2+y^2+z^2\geq1}\frac{e^{\sin(x+y+z)}}{(x^2+y^2+z^2)^p}$$ in dependence of $p$. In class we did ...
0
votes
1answer
23 views

Computing taylor series for two variables

Question: Compute all the terms in the taylor series for the following function around the point $(1,1)$ $$f(x,y) = x^2 + y^2$$ My attempt: $$f_x= 2x, \ f_y= 2y, \ f_{xx}= 2, \ f_{yy}=2$$ So we ...
0
votes
1answer
18 views

Find the volume of the region outside cone and inside sphere.

Find the volume of region outside the cone $\varphi = \frac{\pi}{4}$ and inside the sphere $\rho =4cos(\varphi)$. Solution Attempt: I can visualize the surfaces and see that the volume is two ...
1
vote
2answers
41 views

Proving no vector potential for gravitation field defined on all of $\mathbb{R}^3 -$ origin

Let: $$F=\frac{x,y,z}{(x^2+y^2+z^2)^{3/2}}$$ Show that there is no vector potential for F which is defined on all of $\mathbb{R}^3 - \text{origin}$ I can find a vector potential which is not ...
5
votes
1answer
49 views

Prove that $\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}}$ does not exist

Given $\frac{a}{c} + \frac{b}{d} = 1$ Prove that $$\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}}$$ does not exist. So I have done the proof for strict inequalities. And the ...
0
votes
0answers
17 views

flux of a vector field on the surface of a sphere

I attached a picture of the question, but basically have to find the flux of a field on the surface of a sphere. Ive tried the divergence theorem but it doesnt seem to be working.
1
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0answers
13 views

Calculating the lipschitz constant of this function?

So I have a function $f:\mathbb{R}^n \times \mathbb{R}^m \mapsto \mathbb{R}$ given by $$f(x,y) = \sum_{i=1}^m y_i A_ix$$ where $A_i$ are $n$ by $n$ real symmetric positive definite matrices (not that ...
0
votes
1answer
48 views

Finding A 1-form on $R^2 - {\{(0,0)}\}$

I want to find a 1-form on $R^2 - {\{(0,0)}\}$ such that $w(Y) = 0$ and $w(X) = 1$. Here, $$X = -y\frac{\partial }{\partial x} + x\frac{\partial}{\partial y}\ \text{and}\ Y = x\frac{\partial ...
0
votes
1answer
26 views

Proving functions are in neighborhoods and their partial derivatives

Let $g:\mathbb{R}_+\rightarrow\mathbb{R}$ is smooth, $g(1)=1$, and $g'(1)\ne \space \lambda$ (a)Prove that the set $S:=\{(x,y,z)\in \mathbb{R}_+^3 \vert \space x+g(x)+y+g(y)+z+g(z=6)\}$ is locally, ...
0
votes
0answers
15 views

Computing the line integral using stokes theorem

Question $\oint_C \mathbf{F}\bullet d\mathbf{r}$ where $$\mathbf{F}(x,y,z)=(y+z,z+x,x+y)$$ where C is the intersection of the cylinder $x^2 + y^2 = 2y$ and the plane $y = z$ My attempt: Am I correct ...
2
votes
1answer
576 views

Triple Integral of region bounded by cylinder $x^2 + 3z^2 = 9$ and the planes $y = 0$ and $x + y = 3$

Here is the questoin with a diagram. My attempt at solution: $$x^2 + 3z^2 = 9 \Rightarrow 3z^z = 9-x^2 \Rightarrow z^2 = 3 - \frac{x^2}{3} $$ $$\Rightarrow -\sqrt{3 - \frac{x^2}{3}} \leq z \leq ...
0
votes
1answer
24 views

Chain rule for $f(X(t), Y(t))$ where $X, Y : R \to R^2$

I'm having some trouble on understanding how to calculate the derivative of $g(t)$ with regards to $t$, where $g(t) := f(X(t), Y(t))$ and $X(t)$ and $Y(t)$ are $2d$ vectors. That is $X,Y: R \to R^2$. ...
0
votes
0answers
23 views

Multivar Calc. - Bounds of Integration for Change of Variables

I'm studying for my final and I'm a little confused on getting the bounds of integration when you change variables (Transformation from $(x, y) \to (u, v))$. The problem I came across is: $x = u^2 + ...
0
votes
2answers
38 views

Finding critical points for a trigonometric function

Question Calculate the critical point(s) for the following function $$f =\sin(x)+\cos(y) +\cos(x-y)$$ Hint: Use the trigonometric identity $$\sin(\alpha) +\sin(\beta) = ...
14
votes
8answers
2k views

How can I find 3 positive numbers that have a sum of 1 and the sum of their squares is minimum?

How can I find 3 positive numbers that have a sum of 1 and the sum of their squares is minimum? So far I have: $$x+y+z=1$$ $$z=1-(x+y)$$ $$f(x,y)=xyz=xy(1-x-y)$$ But I'm stuck from here. Hints?
1
vote
2answers
39 views

Calculate the line integral of a half circle as a standing unit circle?

Calculate the line integral $$ \rm I=\int_{C}\mathbf{v}\cdot d\mathbf{r} \tag{01} $$ where $$ \mathbf{v}\left(x,y\right)=y\mathbf{i}+\left(-x\right)\mathbf{j} \tag{02} $$ and $C$ is the semicircle of ...
6
votes
1answer
453 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
3
votes
4answers
57 views

Is polar coordinates enough to prove that a limit exists

Somewhat of a basic question but I failed to find an answer or come up with a formal one myself. Suppose I want to find the limit $\lim_{{(x,y)} \to {(0,0)}}f(x,y)$ using spherical coordinates ...
0
votes
0answers
16 views

Find $\alpha, \beta$ s.t. the following is minimized

Hello I would like to find $\alpha,\beta$ s.t. $$ e(\alpha,\beta) = ||\sqrt{1+\gamma^2}-\alpha-\beta\gamma||_\infty = || f_{\alpha,\beta}||_{\infty} $$ Is minimum (consider $\gamma \in [0,1)$, ...