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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0answers
23 views

Problem 10 chapter 9 from PMA Rudin

If $f$ is a real function defined in a convex open set $E\subset \mathbb{R}^n$, such that $(D_1f)(\mathbf{x})=0$ for every $\mathbf{x}\in E$, prove that $f(\mathbf{x})$ depends only on $x_2, \dots, ...
2
votes
1answer
27 views

Basis representation of differential-form $f(x_1,x_2):=(x_1+x_2, -x_1, -x_2)^T$

I am trying to learn differential forms. I have read some scripts about differential forms and now I am trying to solve some problems. So the problem is: given $f: \mathbb{R}^2 \to \mathbb{R}^3, ...
1
vote
1answer
36 views

Find the partial derivative of a sphere with equation $x^2+y^2+z^2=4$

We have a sphere with the following equation: $x^2+y^2+z^2=4$ We seek to find the partial derivative, with respect to $x$, of this equation. We think of this equation as a function of three ...
15
votes
2answers
400 views

Iterated Integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac1{1-xyuv}\,dx\,dy\,du\,dv$

In page 122 of a book by William J. LeVeque, indeed Topics in Number Theory (1956), there is an exercise for evaluating the following integral in two ways. $$\int_0^1\int_0^1\frac1{1-xy}\,dx\,dy$$ ...
0
votes
1answer
18 views

Finding the center of mass for a centroid without a convenient symmetry axis

Find the centroid of the lamina described in polar coordinates as $\left \{ \strut \left ( x,y \right )~|~0\leq r\leq 4 \cos\left ( \theta \right ),0\leq \theta \leq \frac{\pi}{3} \right \}$ Having ...
0
votes
2answers
49 views

Evaluate the limit or prove that it does not exist [on hold]

I want to evaluate $\displaystyle \lim_{(x,y)\to (0,0)}\frac{\ln(1-x^2-y^2)}{x^2+y^2}$. Any idea how to prove the answer is -1? I don't see an easy way to simplify this.
0
votes
1answer
15 views

How to show map is non-singular

Let $f:\;\mathbb{R}^n\to\mathbb{R}^n$ be differentiable. Suppose that for all $x\in\mathbb{R}^n:$ $$\lVert \mathrm{D}f(x)-\mathrm{I}\rVert\leq \frac{1}{2}$$ where $\lVert\cdot\rVert$ is the ...
3
votes
1answer
86 views

Hessian matrix of $g\circ f$

Say, $f:\mathbb R^n\to\mathbb R^k$ and $g:\mathbb R^k\to\mathbb R$ are both $C^2$. I'd like to express the Hessian matrix of $g\circ f$ $$\left( \frac{\partial^2(g\circ f)}{\partial x_i \partial ...
0
votes
0answers
34 views

Show $\frac{y-x}{(2-x-y)^3}$ is not integrable on $[0,1]\times[0,1]$, not invoking Fubini's theorem.

The double integral $$I = \int_{[0,1]\times[0,1]}\frac{y-x}{(2-x-y)^3} dxdy$$ does not have a finite value. The two iterated integrals have different values (Counterexample to Fubini?). Then Fubini's ...
2
votes
2answers
411 views

How to prove mathematically that two planes parallel to a third plane are parallel

Without relying on geometrical intuition and purely using vector calculus, how do we show that two planes parallel to a third plane are parallel? I assume three dimensional space.
0
votes
1answer
29 views

Derivative of an integral on a level set

Consider a mapping $\xi:\mathbb{R}^d\rightarrow\mathbb{R}^k$ such that $D\xi \, D\xi^T>\delta\, I_k$. Here $D\xi:\mathbb{R}^d\rightarrow \mathbb{R}^{k\times k}$ is the Jacobian. Consider a ...
0
votes
1answer
44 views

Multivariable Calculus, Parametrization and extreme values

I want to find the extreme values of the function $f(x,y,z) = 2x + 2y + z$ under the constraints $x^2+y^2+z^2 \le 2$ and $x^2 + y^2 \le z$ The task is to use a parametrization of the two ...
-1
votes
1answer
40 views

Parametrization of two curves. [on hold]

I have an assigment to parametrize the edge of the volume which is given by the intersection of the two curves $x^2+y^2+z^2=2$ and $z=x^2+y^2$. I really have no idea how i can parametrize this? I know ...
0
votes
0answers
15 views

Derivative of equation in matrix form

I need to compute first derivatives of the following function $S(w)$ with respect to $w$. Then solve it. The reason behind that is to minimize $S(w)$. $S(w)=\sum_{i=1}^{n} w_i^{1/2} \bigg(y_i - ...
2
votes
1answer
455 views

Vector Calculus (Gradients, Potential Functions, and Equipotential Curves)

This is my first question on the Mathematics section of StackExchange, so please forgive me if I don't follow all the rules or things like that. Here's my question: Consider the following potential ...
2
votes
3answers
4k views

Curl of Cross Product of Two Vectors

I want to prove the following identity $$\text{curl } \left(\textbf{F}\times \textbf{G}\right) = \textbf{F}\text{ div}\textbf{ G}- \textbf{G}\text{ div}\textbf{ F}+ \left(\textbf{G}\cdot \nabla ...
0
votes
0answers
11 views

Transforming UV Region to XY Bounded By Hyperbolas and Lines

Suppose I have a region in the x-y plane bounded by: $y=\frac{1}{x}, y=\frac{4}{x}, y=x, y=4x$ We see that: $1\leq yx \leq 4$, and $1\leq \frac{y}{x} \leq 4$ If I let $u=yx$ and $v = ...
1
vote
1answer
16 views

Let $n>1$ and $g_1,…,g_{n-1}$ be $C^2$ scalar fields over $\mathbb R^n$ , then for any scalar field $f$ , is $\det J(f,g_1,…,g_{n-1})=0$?

Let $n>1$ and $g_i:\mathbb R^n \to \mathbb R$ be scalar field for each $1\le i\le n-1$ such that all second order partial derivatives of each $g_i$ exist and are continuous ( i.e. each $g_i$ is ...
14
votes
1answer
298 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
0
votes
1answer
701 views

gradient of vector 2-norm

I have a function $f(\Theta) = \frac{1}{2N}\| y-\mathcal{X}(\Theta)\|_2^2$. Matrix $\Theta\in\mathbb{R}^{m_1\times m_2}$, $y=[y_1,\cdots,y_N]^T\in\mathbb{R}^N$ is the observation vector, and we use ...
0
votes
2answers
26 views

Proving that $f^2$ is differentiable given that f is differentiable at $(x_0,y_0)$

So I've tried using the definition: $f$ is differentiable at $(x_0,y_0)$ iff $$ f(x,y)-f(x_0,y_0)=\frac{\partial f}{\partial x}(x_0)\cdot x+\frac{\partial f}{\partial y}(y_0)\cdot y+o(\sqrt ...
1
vote
0answers
36 views

Fermat's theorem for $\mathbb{R}^n$

Suppose that $f$ is a differentiable function in an open set $E\subset \mathbb{R}^n$, and that $f$ has a local maximum at a point $\mathbf{x}\in E$. Prove that $f'(\mathbf{x})=0.$ I am sorry if this ...
2
votes
1answer
38 views

gradient of gradient is it Hessian?

Say, I have a function $f(\vec{x}) = \cfrac{1}{2}\vec{x}^{T}Q\vec{x} - \vec{b}^T\vec{x}$, where $Q$ is Symmetric Positive Definite $\in R^{nxn}$. I want to find $\nabla f(\vec{x} - \nabla ...
0
votes
0answers
16 views

Change of variables to polar in higher dimensions

For $x_0 \in \mathbb{R}^n$ I'm trying to apply a polar change of variables to write $$ \int_{|x - x_0| < R_0 - c_2t} f(x,t) \, dx = \int_{r=0}^{r=R_0} \int_{S^{n-1}} f(x_0 + r\omega, t)r^{n-1}\, ...
1
vote
0answers
16 views

Differentiation with integration region depending on $x$ to solve for decreasing energy of wave equation

I want to show that for the general wave equation $u_{tt} - \nabla \cdot (c^2\nabla u) + qu = 0, \quad u(x, 0) = \phi(x), \quad u_t(x, 0) = \phi(x)$ we have $$ E(t) = \int_{|x-x_0| < R_0 - c_2t} ...
1
vote
2answers
15 views

Using double integration to compute an average yields a different result than computing it without integration

I'm self-learning multivariable calculus and am using double integration to compute the average value of $f(x,y)$ over some region. I'm trying to solve the following simple problem using two different ...
1
vote
2answers
4k views

Finding the equation of a one sheeted hyperboloid

Was working on calc 3 homework assignment and couldn't find out how to solve this question. the answer doesn't matter any more,i just really want to find out how to do it since my book seems to skip ...
3
votes
1answer
898 views

Proof that the product of two differentiable functions is differentiable

Let $f:A\subset \mathbb{R^p}\rightarrow\mathbb{R^q}$ and $\phi:A\subset \mathbb{R^p}\rightarrow\mathbb{R}$ differentiable in $c\in A$. I have to prove that $g(x)=\phi (x)f(x)$ is differentiable, ...
4
votes
1answer
69 views
+250

Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range

I was studying the Inverse function theorem when I came across the following problems : (Let the closed set $V$ i.e the range have non-empty interior) Does there exist a continuous onto ...
1
vote
1answer
42 views

solve $\frac{\partial u^2}{\partial x\partial y}=0$

I need to solve $$\frac{\partial u^2}{\partial x\partial y}=0$$ with the boundary conditions: $u(x,y=x^3)=\sin(x^6)$ and $\frac{\partial u}{\partial x}(x,y=x^3)=0$. I got a particular solution, I ...
2
votes
4answers
17k views

Finding parametric equations for the tangent line at a point on a curve

Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$ I understand that in ...
0
votes
2answers
25 views

Find $f(\Bbb R^2)$ where $f(x,y) = (e^x \cos y, e^x \sin y)$

Let $f(x,y) = (e^x \cos y, e^x \sin y)$. What is $f(\Bbb R^2)$? I know I should take $u = e^x \sin y$ and $v =e^x \cos y$ and try to find a relation but I can't find something
0
votes
1answer
13 views

Taking derivative of energy of wave equation

Consider the variable coefficient, real valued wave equation $$ u_{tt} - \nabla \cdot (c^2 \nabla u) + qu = 0, \quad u(x,0) = \phi(x), \quad u_t(x, 0) = \phi(x), $$ where $c, q \geq 0$ depend only on ...
1
vote
1answer
532 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
-2
votes
1answer
29 views

Infimum and supremum of two variable function [on hold]

How can I find the infimum and supremum in $\mathbb{R}^{2} $ of this function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$? Thanks EDIT: Forgive me if I did not add my thoughts but I did not know where to ...
2
votes
1answer
41 views

Definition of partial derivatives from Rudin's PMA

It's the definition of partial derivative from Rudin's PMA. Why he consider $(25)$ for real functions $f_i$? What about if $f_i$ in $(25)$ replaced by vector-valued function ...
0
votes
1answer
29 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
0
votes
0answers
17 views

Negative eigenvalue of a hessian matrix entails a local decrease in function value?

I was reading up on non-convex optimization, and I can across this sentence: "Since Hessian(f(w)) has a negative eigenvalue, there is always a point that is near w which has smaller function value" ...
-4
votes
0answers
35 views

How to prove the following questions by IBP? (Integrated By Parts) [on hold]

So this is the question that I have to solve. I know this is related to IBP, but Have no idea how to start and prove... need help
0
votes
1answer
45 views

Graph of $f(x,y) = \frac{3x^2 y}{x^2+y^2}$ near the origin

I am trying to graph the function $f : (x,y) \mapsto \frac{3x^2 y}{x^2+y^2}$ on a TI-89 Titanium. I have noticed that no matter how many times I zoom in toward the origin the graph appears identical. ...
1
vote
1answer
37 views
+50

Complicated surface integral/line integral.

Problem Compute the integrals $$I=\iint_\Sigma \nabla\times\mathbf F\cdot d\,\bf\Sigma$$ And $$J=\oint_{\partial\Sigma}\mathbf F\cdot d\bf r$$ For $F=(x^2y,3x^3z,yz^3)$, and ...
0
votes
2answers
45 views

Method of characteristics - finding the particular solution using initial conditions

I am trying to use the method from my previous question to solve this PDE: $$ 3u_x + 2u_t = \cos x $$ with initial condition $u(x,0) = x^2$. So I need to solve these: \begin{align} \frac{dx}{ds} ...
2
votes
0answers
25 views

singular $1$ cube - Boundary of $2$ chain

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
3
votes
1answer
181 views
+50

Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

Consider the following boundary value problem (BVP) $$\matrix{ {{\Delta ^2}H = 0,} \hfill & {} \hfill & {{\rm{in}}\,} \hfill & \Omega \hfill \cr {\partial _y^2H = 0} \hfill & ...
3
votes
0answers
90 views
+50

A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
0
votes
1answer
28 views

Finding correct variation for $\rho$ in spherical coordinate integration

I am having some trouble and looking for help on calculating the moment of inertia about the z axis of the region bound by the cone $z=\sqrt{3(x^2+y^2)}$ and the sphere $x^2+y^2+z^2=a^2$ if the ...
2
votes
1answer
2k views

Does the Divergence Theorem Work on a Surface?

The divergence theorem in $\mathbb{R}^3$ says that the integral of the divergence of a vector field over a solid $\Omega$ in $\mathbb{R}^3$ equals the flux through the surface of $\Omega$ denoted by ...
0
votes
1answer
25 views

Differentiating a function composition

Given $g:R^n \rightarrow R^k$ and $h:R^k \rightarrow R$, we have $f(x) = h(g(x))$. Using the chain rule, we can differentiate $f(x)$ to get $f'(x) = \nabla^Th(g(x))g'(x)$ My question is why do we ...
4
votes
1answer
67 views

Integral of an unbounded function as a solution of $\nabla^2\boldsymbol{A}=-\boldsymbol{J}$

While studying the equivalence between the Biot-Savart and Ampère's laws I have only found proofs of the fact that$$\boldsymbol{A}(\boldsymbol{x})=\frac{\mu_0}{4\pi}\int_V ...
2
votes
3answers
60 views

Why doesn't the limit $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ exist?

Why is this limit non-existant? $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ I can't seem to find $2$ different paths that would show it is non-existant.