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Questions tagged [multivariable-calculus]

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).

2
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1answer
44 views

Help with the notation $(x,t)\in \mathbb R^n \times (0,\infty)$

What is the meaning of $$(x,t)\in \mathbb R^n \times (0,\infty)\quad ?\tag 1\label1$$ I guess $x$ is a $n$-vector and $t$ is just a scalar, i.e. \begin{align} x&=(x_1, x_2, \dots, x_n)\in \...
0
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1answer
17 views

Question regarding complex differentiability and vector differentiability from $R \rightarrow R^2$

Suppose I have a function defined from R to C: $f(t)=R(t)+iI(t)$ When someone wants to show that the above function is differentiable are they talking about complex differentiability or vector ...
1
vote
1answer
25 views

Show that $f$ is of class $C^{k}$ and compute $f'$

Let $D$ be an open set in $\Bbb{R}^n$ and $$u :D\to L(\Bbb{R}^n,\Bbb{R}^n)$$ be a function of class $C^k$ for $k\geq 1.$ Let $$f :D\to \Bbb{R}^n$$ be defined by $f(x)=u(x)(x).$ I want to show that ...
0
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2answers
38 views

Transform the differential equation $yu_x(x, y) - xu_y(x, y) = xyu(x, y)$ by introducing new variables $s = x^2+y^2$ and $t = e^{-x^2/2}$

Transform the differential equation $$yu_x(x, y) - xu_y(x, y) = xyu(x, y)$$ by introducing new variables $s = x^2+y^2$ and $t = e^{-x^2/2}$. Then determine the general solution to the equation. ...
0
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1answer
22 views

Show that $\varphi$ is of class $C^{\infty}$ and compute $\varphi'$

I am trying to solve the following question, arising from Implicit Function Theorem. $$\varphi :L(\Bbb{R}^n,\Bbb{R}^n)\to L(\Bbb{R}^n,\Bbb{R}^n)$$ $$A\mapsto\varphi(A)=A^2.$$ I want to show that $\...
1
vote
1answer
17 views

Derive the gradient of the second order Taylor approximation

I study second year applied physics and electrical engineering and I am preparing for an exam in optimization. I am trying to understand the derivation/calculation of the step direction $\mathbf{d}$, ...
1
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3answers
60 views

If $\Omega\subset\mathbb{R}^n$ is convex and $f$ is differentiable, then $f(x)-f(y)\geq f'(y)(x-y),\;\forall \;x,y\in \Omega$

Let $\Omega$ be a convex set in $\Bbb{R}^n$. We say that that $f:\Omega\to \Bbb{R}$ is convex if $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y),\;\forall\;0\leq t\leq 1,\;\&\;\forall\;x,y\in \Omega.$$ I want ...
1
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1answer
32 views

Divergence theorem and a hemisphere

I'm confused about applying the Divergence theorem to hemispheres. Here is the statement: As far as I understand, this question asks to compute $\int\int_{S_1}\mathbf F\cdot d\mathbf S$ over $$S_1=\...
0
votes
0answers
31 views

Proof of partition of unity in Spivak's Calculus on manifolds

In Case 3 of Spivak's proof of partition of unity, he seems to assume the existence of boundary of $A$. I wonder if there is a case where the boundary $A$ is $\Phi$, which requires an extra proof? ...
5
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3answers
59 views

Edit: Show that $f$ is differentiable on $\Bbb{R}^n$ and compute $f'$

Good day all! Edit: I'm currently doing a personal study on differentiation on $\Bbb{R}^n$ but I have this challenging problem. Although, some answers have been provided on how to show that show that ...
1
vote
1answer
19 views

Different results after changing the order of integration with constant limits (Failure of Fubini's theorem)

I have the following question $I_{1}=\int _{0}^{1}\int _{0}^{1}\ \frac{(x-y)}{(x+y)^{3}}\ dy\,dx$ Evaulating the above I get $I_{1}=0.5$ Now if I switch the order of integration $I_{2}=\int _{0}^{...
0
votes
1answer
54 views

Evaluating $\oint_C (y\;dx+z\;dy+x\;dz)$ [on hold]

Let $C$ be the curve resulting from the intersection of surfaces $x+y+z=0$ and $x^2+y^2+z^2=4$. Then what is the value of $\oint_C (y\;dx+z\;dy+x\;dz)$? What I obtained is $12\pi/\sqrt{3}$ while ...
0
votes
1answer
44 views

Use the implicit function theorem to prove that $f=f^{-1}$.

I've had some problems to prove this proposition: Let $f:\mathbb{R}^{n}\to \mathbb{R}^{n}$ a function of class $C^1$ such that $(f\circ f)(x_0)=x_0$ for some $x_0\in \mathbb{R}^{n}$ and $D(D(f(x)))=...
1
vote
1answer
54 views

How to compute $\int_H F\cdot n \, dS$ efficiently?

Let $H=\{(x,y,z):z>0, x^2+y^2+z^2=R^2\}$ and $$F(x,y,z)=(x^2(y^2-z^3),xzy^4+e^{-x^2}y^4+y,x^2y(y^2x^3+3)z+e^{-x^2-y^2})$$ Find $\int_H F\cdot n\,dS$ where $n$ is the outward unit normal and $dS$ ...
0
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1answer
17 views

If $A$ is convex and $f$ has first partial derivative zero, then $f(x',y) = f(x,y)$ for every $(x',y),(x,y) \in A$?

Let $A \subset \mathbb{R}^{2}$ be open and $f:A \to \mathbb{R}$ of class $C^{1}$. Suppose that $D_{1}f(a) = 0$ for every $a \in A$. (a) If $A$ is convex, then $f(x',y) = f(x,y)$ for every $(x',y),...
0
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0answers
20 views

Why does a new restriction suddenly appear when finding this potential function using antiderivatives?

I have 3 questions regarding the solution to the following problem: Let $$\vec{F} = M \hat i + N \, \hat j=\big(\sqrt{x^2+y^2}\big)^n (x \, \hat i + y \, \hat j)$$Whenever possible, find a ...
1
vote
3answers
40 views

How to evaluate $\int_S(x^4+y^4+z^4) \, dS$ over surface of the unit sphere.

Question. Let $S$ denote the unit sphere in $\mathbb{R}^3$. Evaluate: $$\int_S (x^4+y^4+z^4) \, dS$$ My Solution. First I parametrize $S$ by $$r(u,v)=(\cos v \cos u, \cos v \sin u, \sin v)$$ $0\le u \...
2
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0answers
40 views

If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $

Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...
1
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0answers
26 views

Measure of $f^{-1}(y)$ is zero

Let $I=[0,1]^d$ be the $d$-dimensional unit box, let $f\in C^1(I)$ a real-valued function, and let $y\in \mathbb{R}$ be in the image of $f$. My question: What will be a "miniml" sufficient condition ...
4
votes
1answer
46 views

Prove that if $\langle df_x(v),v\rangle>0$, then $f$ is injective

Suppose $f:\mathbb R^n\to \mathbb R^n$ is a $\mathcal C^1$ map such that $\langle df_x(v),v\rangle>0$ for all $x\in \mathbb R^n$ and $v\in R^n\setminus \{0\}$. Prove that $f$ is injective. Hint: ...
1
vote
1answer
25 views

Prove that the intersection of two surfaces can be expressed as a curve of a given form

The point $(1,-1,2)$ lies on each of the surfaces $$x^2(y^2+z^2)=5,\ (x-z)^2+y^2=2.$$ Prove that in a neighborhood of that point, the intersection of the surfaces can be described as a smooth curve $z=...
2
votes
1answer
60 views

On line integrals of $\frac{xdy-ydx}{x^2 +y^2}$

There are a few questions on MSE about integrals of the form $$\int_C \frac{xdy-ydx}{x^2 +y^2},$$ where $C$ is a smooth simple closed positively oriented curve; but none of them gave me a complete ...
1
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0answers
28 views

Using taylor expansion to show the negative of the gradient gives the most rapid decrease

This is an explanation of why $-\nabla f_k$ is the better direction to step down. I undrstand why $-\nabla f_k$ is the direction that minimizes $p^t\nabla f_k$, why minimizing $p^T\nabla f_k$ will ...
2
votes
1answer
27 views

Volume enclosed by paraboloid and plane

The volume, $V$, enclosed by paraboloid $z=x^2 + y^2$ and the plane $z=3-2y$ can be expressed as a triple integral. Determine the limits describing the enclosed volume. By evaluating the integral, ...
0
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0answers
31 views

Optimization problem over integration sublevel set in $\mathbb{R}^n$

Suppose $\phi:\mathbb R^n\to\mathbb R^n$ is a fixed smooth vector field; if it's useful, we can assume $\|\phi(x)\|_2\leq1$ for all $x\in\mathbb R^n$. Consider the following optimization problem ...
1
vote
1answer
46 views

Double Integral over Trapezium

$\int \int_ A \sin\left(\frac{y-x}{x+y}\right)dA$ where $A$ is the trapzeium with vertices $(1,1), (2,2), (2,0), (4,0)$. I decided to use $u=y-x$ and $v=y+x$ as my variables, and from attempting to ...
1
vote
1answer
35 views

Prove that there is a solution that depends smoothly on the coefficients

Consider the equation $x^4+a_0x^3+a_1x^2+2a_2x+a_3=0$. Prove that there is $\delta > 0$ such that if $|a_i-1| < \delta$ for $i=0,1,2,3$, then the above equation has a solution that depends ...
0
votes
1answer
22 views

Show that a proper locally invertible map is surjective

Let $f:R^2\to R^2$ be a continuously differentiable function such that $Df(x)$ is invertible for all $x\in R^2$ and $f^{-1}(K)$ is compact for every compact set $K$. Show that $f$ is surjective. The ...
2
votes
0answers
29 views

Find $f$ for which there is a vector field $W$ with $\operatorname{curl}W=(y,x,f(x,y,z))$

For what smooth functions $f:\mathbb R^3\to \mathbb R$ is there a smooth vector field $W:\mathbb R^3\to \mathbb R^3$ with $\operatorname{curl}W=V$, where $$V(x,y,z)=(y,x,f(x,y,z)).$$ For $f$ in this ...
0
votes
2answers
41 views

Change the integral into polar coordinates

Convert the integral $\int_{-2}^{2} \int_0^{\sqrt{4-x^2}} e^{-(x^2+y^2)} dy\,dx$ into polar coordinates don't need to solve. my solution is currently $\int_0^2 \int_0^\pi re^{-r^2} d\theta\, dr$ not ...
0
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0answers
29 views

Cauchy-Riemann Reparametizing level curves to gradient flow lines Using Picard-Lindelof

Problems in question here How can I prove that level curves of function V can be reparametrized to gradient flow lines of U? Any help would be appreciated and if anybody has any tips for part 3 would ...
3
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0answers
53 views
+50

Intuition or motivation for the definition of an hypersurface. What are we actually trying to define?

If we have $x^2 + y^2= 1 $ then we can solve for $y$ and $x$, at least in parts. The implicit function theorem gives us the conditions to solve these things. At this part of this book (Folland, about ...
1
vote
0answers
47 views

Prove that $\det Df\equiv 0$ if $g\circ f \equiv 0$

Let $f:R^n\to R^n$ be a smooth function and $g:R^n\to R$ $$g(x_1,\dots,g_n)=x_1^5+\dots+x_n^5$$ Assume that $g\circ f\equiv 0$. Prove that $\det Df\equiv 0$. The only idea I have is to apply the ...
0
votes
3answers
49 views

Find the maximum of $f(x, y) = (3x+4y)e^{\frac{-(x^2+y^2)}{2}}$ as a function of $r$ over a closed disk $x^2+y^2\leq r^2.$

Find the maximum of $f(x, y) = (3x+4y)e^{\frac{-(x^2+y^2)}{2}}$ as a function of $r$ over a closed disk $x^2+y^2\leq r^2.$ So my first guess is to check the interior points and then the boundry ...
2
votes
2answers
61 views

Deriving the Taylor expansion $f(x+p) = f(x) + \nabla f(x+tp)^Tp$

I'm trying to derive the Taylor formula: $$f(x+p) = f(x) + \nabla f(x+tp)^Tp$$ For that I think tha I just need to use the formula for one variable taylor expansion and follow like here: https://...
0
votes
0answers
24 views

question regarding Inverse function theorem

I was reading Munkres's Analysis of Manifolds. I got stuck on a lemma. I am stating this lemma here. Let $A$ be an open set in $R^n$ and let $f:A \to \Bbb{R}^n$ be of class $C^1$. If $Df(a)$ is non-...
3
votes
1answer
45 views
+50

Understanding this very generic divergence theorem where the open set have border $C^k$

I'm at a PDE class and my teacher gave a very generic definition of the divergence theorem. I can't find it anywhere. It's something like this: Definition: let $k\in \{1,2,\cdots,\infty\}$, $N\ge 2$ ...
-1
votes
1answer
54 views

How important is the assumption $\gamma$ is positively oriented? (Residues, Cauchy's Thm from Cauchy's Integral Formula)

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 4.32, Cor 8.27 Question 1. Should the following 2 statements in the textbook have an ...
0
votes
1answer
22 views

Uniqueness of the maximum of a multi-dimensional function

I have a somewhat complicated function of $M+1$ variables, which looks as follows. $$f (x_0, x_1, x_2, \dots, x_M) = \sum_{i=1}^{N_A} \ln \left[1 - \text{erf}\left(x_0 + \sum_{j=1}^M a_{ij} x_j\right)...
3
votes
2answers
48 views

$f:\mathbb{R}^4 \to \mathbb{R}$ is a Constant when $Xf = Yf = 0$ for Vector Fields $X,Y$

I'm preparing for some comprehensive exams and this is a question from a previous year that I've been trying to solve. "On $\mathbb{R}^4$, equipped with coordinates $(x,y,z,t)$, let $X,Y$ be vector ...
5
votes
0answers
66 views

Triangle inequality for integrals, but for an arbitrary norm

Given an arbitrary norm on $\mathbb{R}^q$. For a continous function $f: [a,b]\times\mathbb{R}^q \rightarrow \mathbb{R}^q$, I want to find out whether $$ \left\|\int_{a}^{b} f(x,y_1,...,y_q)\,\mathrm{...
2
votes
1answer
53 views

Finding center of a solid

I am finding the center of mass of sphere $x^2+y^2+z^2=2z$ when $\delta(x,y,z)=\sqrt{x^2+y^2+z^2}$. I did the mass as follows (I hope it is right): $$M=\int_0^{2\pi}d\theta\int_0^{\pi/2}\int_0^{2\cos{...
-1
votes
2answers
39 views

Determine the nature of a critical point

I'm asked to determine the nature of the critical point f(0,0), but the second derivative test fails here, how can I solve this? Below is the function of which we are asked to find the extrema and ...
3
votes
2answers
83 views

Bounding $(x^n-y^n)/(x-y)$?

Let $x,y \in [0,1]$ it follows then from the binomial theorem that for integers $n \ge 1:$ $$\sup_{x,y} \left\lvert \frac{x^n-y^n}{x-y} \right\rvert \le n.$$ Is this also true if $q \in [1,\infty)$:...
2
votes
2answers
297 views

Domain of multivariable function

I have a function of two real variables which is given by the transformation rule $$f(x,y)=\frac{y}{1+x^2+y^2}.$$ I have to find the domain of $f$ which consists of all points $(x,y)$. When I examine ...
1
vote
1answer
34 views

If f is differentiable in a region $G$, then f is infinitely differentiable in G, and all partials of $f$ are continuous

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch 5.1, Exer 4.25 (Cor 5.5) If f is differentiable in a region $G$, then f is infinitely ...
-2
votes
2answers
59 views

Weaker condition for path independence of $\int_{\gamma } f$ on disconnected open subset

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Cor 5.9 (Cor 5.9) If $f$ is a holomorphic function on a simply-connected region $G \subseteq \...
2
votes
1answer
55 views

What is the point of computing unit normal vector before computing image under stereographic projection?

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.27 (Exer 3.27) Consider the plane $H$ determined by the equation $x + y -z = 0$. ...
0
votes
1answer
20 views

Function where for which any change in xy-plane implies change in z

I'm walking along the surface of some function f(x,y)=z. Now I come to a point (x0,y0,z0) from which I cannot move continuously to another point with the same z-coordinate, i.e., some (r,s,z0) where (...
0
votes
2answers
37 views

Proof of $\nabla f(x^*) = 0$ is necessary condition for minimizer, using taylor expansions

Look at this part: Define the vector $p = -\nabla f(x^*)$ and note that $p^T\nabla f(x^*) = -||\nabla f(x^*)||^2 <0$. Because $f$ is continuous near $x^*$, there is a scalar $T>0$ such that ...