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
13 views

Representation of derivative by alinear map

Let $\phi:\mathbb R^2 \to \mathbb C$ be the map $\phi(x,y)=z$ where $z=x+iy$.Let $f:\mathbb C\to \mathbb C$ be the function that is $f(z)=z^2$and $F=\phi^{-1}f\phi $.. Represent the derivative of ...
3
votes
1answer
18 views

Finding the parameterization of a curve for a line integral problem

I have to calculate the work of a particle that travel along a curve, given the following vector field: $F(x, y, z) = (2z-1, 0, 2y)$ and where the curve is the intersection between: $s1: z = x^2 + ...
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3answers
48 views

Tangent to surface given a point [on hold]

so is the way I approach this : gradF(x,y,z)?
0
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0answers
19 views

Computing gradient using implicit functions theorem

I am trying to compute analytically the gradient of a function to specify it in a Python program and find the minimum more rapidly. I did the computation many many times and I do not manage to find ...
0
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2answers
25 views

evaluating the double integral

I tried to calculate $\int _0^9 dx\:\int _{-\sqrt{x}}^{\sqrt{x}}\:y^2dy$ which yielded $c$ as in this integral has no particular value...when I plot the graphs for it's D however, a certain area does ...
2
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2answers
27 views

Proving Hadamard lemma: how to apply FTC in first step?

I wanted to prove Hadamard's lemma but got stuck on the first step: Let $f \in C^\infty (\mathbb R^n)$ and $x_0 \in \mathbb R^n$. Then there exist $g_i \in C^\infty (\mathbb R^n)$ such that $$ f(x) ...
3
votes
2answers
27 views

Computing line integral using Stokes´theorem

Use Stokes´ theorem to show that $$\int_C ydx+zdy+xdz=\pi a^2\sqrt{3}$$ where $C$ is the curve of intersection of the sphere $x^2+y^2+z^2=a^2$ and the plane $x+y+z=0$ My attempt: By Stokes´ theorem ...
0
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0answers
29 views

Can I apply chain rule for two different equations?

I have two arbitrary functions $F(x,y)$ and $G(x,y)$ My question is pretty simple, say $F$ and $G$ are differentiable for all $x$ and $y$. Taking derivatives of $G$ and $F$ with respect to $x$ and ...
0
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0answers
20 views

joint optimization problem with somewhat symmetric function

I have just brief question that the method that I use to solve optimization problem is legit. I have function $\max_{x,y}F(x,y)$, and first order condition gives me following equation. ...
1
vote
1answer
21 views

Total differential of composite function

Let function $f(x,y)$ is defined at neighborhood of $(1,1)$ and has a continous partial derivations here. Let $g(x,y) = f(f(x,y), f(y,x))$ and we know, that $f(1,1) = 1$, $\partial _x f(1,1) = 1$ and ...
0
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1answer
31 views

Describing a set

Is it possible to describe a set $\left\lbrace(x,y,z)\mid x \sin z + y \cos z = e^z \right\rbrace$ on the neighborhood of $(2, 1, 0)$ as a graph of a function $f$, where $f(2,1) = 0$?
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0answers
23 views

Make this proof rigorous

Theorem. Consider the following differential equation: $$\eqalign{ Q(q,p) &= \frac{dq}{dt} \cr P(q,p) &= \frac{dp}{dt} }$$ Suppose that for any compact measurable region $\Gamma \subset R^2$, ...
0
votes
1answer
25 views

polar coordinates question

I was tasked with writing $\iint_D f(x,y) \,dx \,dy$ for $ [ D:{4\leq x^2 + y^2 \leq}9]$ through ''reoccurring integrals'' in polar and Cartesian systems? what are ''reoccurring integrals''? and how ...
1
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0answers
27 views

polar system in a plane?

what is a polar system in a plane and how it helps in calculating integrals in certain areas? I'm looking for a good explanation/a fair/ readable source on the matter.
1
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2answers
34 views

Change of variables in multivariable differential equations

This is a very easy question about how to justify the change of variables. Let $f$ be a $C^1$ function of two variables $x,y$. Introduce the variables $s,t$ as: $$\begin{cases} s=x+y \\ t=x-y ...
1
vote
1answer
13 views

ratio between volumes in $\mathbb{R}^n$

Let $[-a_n,a_n]^n$ be the largest cube that fits into the n-sphere $S^{n-1}.$ Can we say what $a_n$ is? I mean, for $n=1$ we have $a_1=1$ and for $n=2$ we have $a_2 = \frac{1}{\sqrt{2}},$ so does ...
0
votes
1answer
37 views

What does the region look like?

I am asked to calculate the double integral over the triangle R defined by $-x\tan (\alpha) \le y \le x\tan (\alpha)$ and $x \le 1$. It is also noted that $\alpha$ is an acute angle. I am having a ...
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0answers
27 views

Double Integral to evaluate volume over region [on hold]

I'm not sure how to write the integral needed for this problem: Find the volume of the solid bounded by the graphs of the equations: $$x^2 + z^2 = 1 $$ $$y^2 + z^2 =1 $$ And the first octant. I ...
0
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0answers
24 views

Finding the value of $K$ given a surface integral

Question: If $S$ is the surface of $z^{2} = 1 + x^{2} + y^{2}$ between $z= 1$ and $z = K$ for $K > 1$ and $\int\int_S zdS = \dfrac{4912\pi}{3}$ then what is the value of $K$? Relevant ...
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0answers
7 views

Line integrals in a double connected set

If P and Q are continuously differentiable on an open doubly connected(one hole) region $R$, and if $\partial P/\partial y = \partial Q/\partial x$ everywhere in $R$, how many distinct values are ...
1
vote
1answer
40 views

Calculating the flux

I am a little unsure about setting the boundaries after I set up my equation. The function is given as $F (x,y,z) = (x^{1/3}, - y^{1/3}$, and $2y/(x^{1/3}) )$ The surface it is over is given as ...
1
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0answers
23 views

Multiple Integration of a line and an ellipse.

Let $A$ be the area of the region in the first quadrant bounded by the line $y=\dfrac{1}{2}x$, the x-axis, and the ellipse $\dfrac{1}{9}x^2+y^2=1$. Find the positive number $m$ such that A is equal to ...
-4
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0answers
25 views

Partial derivatives of the exponential-multivariable

Consider the function $u(x,y)=e^{x^4y^6}$. How do I calculate $\frac{\partial^{35}u}{\partial x^{20} \partial y^{15}}(0,0)$ using Taylor series?
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0answers
27 views

Finding bounds of integration

Let $S$ be the region in the first quadrant of the $xy$-plane bounded by the $x$-axis and the parabolas $$x=1-\dfrac{1}{4}y^2,$$ $$x=\dfrac{1}{4}y^2-1$$ and $$x=4-\dfrac{1}{16}y^2.$$ Use the ...
0
votes
2answers
40 views

changing order of integration

I was tasked with changing the order then drawing and calculating the integral $\int _0^2 dy\:\int _{y}^{y+2}\:\frac{x}{y+2}dx$...which got very complex. I understood that the D area had to be split ...
1
vote
4answers
101 views

The integral of Gaussian function of three variables [closed]

How do I solve this $$\displaystyle\int_{-\infty} ^{\infty}\displaystyle\int_{-\infty} ^{\infty}\displaystyle\int_{-\infty} ^{\infty} e^{-x^2-y^2-z^2}\ dx \ dy \ dz$$
3
votes
1answer
104 views

Clarification on Implicit Derivatives steps

I have been attempting to wrap my head around this problem for a couple days now. I've attempted numerous different iterations to try and find how the answer is derived, but I just don't see the ...
3
votes
3answers
73 views

How to prove $\lim\limits_{(x,y)\to(0,0)}\frac{{x^3{y^2}}}{{{x^4} + {3y^4}}} = 0$?

To prove that $$\lim\limits_{(x,y)\to(0,0)}\frac{{x^3{y^2}}}{{{x^4} + {3y^4}}} = 0$$ I start with $$\left| {\frac{{{x^3}{y^2}}}{{{x^4} + 3{y^4}}}} \right| \leqslant \left| ...
1
vote
1answer
57 views

Partial derivatives of $\ln(x^2+y^2)$

I am new to partial derivatives and they seem pretty easy, but I am having trouble with this one: $$\frac{\partial}{\partial x} \ln(x^2+y^2)$$ now if this was just $\frac{d}{dx}\ln(x^2)$ we would get ...
2
votes
2answers
42 views

Limit in multivariable-calculus

Let $\ell$ be a straight line through origo. Determine the limit to the restriction of $$f(x,y)=xye^{-x^2y^2}$$ to $\ell$ when $x^2+y^2 \to \infty$. Also, investigate the limit $$\lim_{x^2+y^2 \to ...
0
votes
1answer
21 views

Prove the Jacobian identity

How to prove that these Jacobians are equal? $$\dfrac{\partial (x,y)}{\partial(\alpha, \beta)} \cdot \dfrac{\partial(\alpha, \beta)}{\partial(z,w)} = \dfrac{\partial (x,y)}{\partial(z,w)}$$ I don't ...
2
votes
4answers
58 views

How can I prove that any ball in $\mathbb{R}^n$ is connected?

As the title follows, how can I prove that any ball in $\mathbb{R}^n$ is connected? or can you give me a hint? I have some ideas but I'm not sure about them. I thank any help you can give me! ...
1
vote
1answer
23 views

Is the set of all $(x,y,z)$ such that $z^2-x^2-y^2-1 = 0$ open or closed?

As the title says, is the set of all $(x,y,z)$ such that $z^2-x^2-y^2-1 = 0$ open or closed? Moreover, how can I prove it? I understand the definition of open and closed sets, but I don't get this ...
0
votes
1answer
8 views

$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty\Rightarrow\lim_{k\to\infty}\nabla f(x^k)=0$

Let $f\in C^1(\mathbb{R}^n)$ and $(x^k)_{k\in\mathbb{N}_0}\subseteq\mathbb{R}^n$ with $$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty$$ Why can we conclude that $$\lim_{k\to\infty}\nabla ...
2
votes
2answers
21 views

Difference between path and vector field

What is the difference between a path and a vector field? From what I understand the unit vectors $\mathbf i$, $\mathbf j$, and $\mathbf k$ are actually vector fields (constant vector fields to ...
4
votes
2answers
113 views

solution of $y^{\prime \prime} + y^n = 1$ [closed]

I am not able to figure out the solution for the differential solution $$y^{\prime \prime} + y^n = 1$$ I want to specifically find an answer for $$y^{\prime \prime} + y^2= 1$$and $$y^{\prime \prime} + ...
0
votes
0answers
14 views

Justifying the “Dual feasibility”, one of the Karush-Kuhn-Tucker conditions

I am having difficulty of interpreting the KKT conditions in a general setting where we have $M$ equality and $N$ inequality constraints defined as: Minimize $f(x)$ subject to $g_i(x) \leq 0 , h_j(x) ...
0
votes
3answers
55 views

Find partial derivatives of $u=x+y+z$, $v=x^2+y^2+z^2$ and $w=x^3+y^3+z^3$

I've been trying to solve this question using the Implicit functions theorem from Schaum's outline series (Theory and Problems of Differential and Integral Calculus, by Frank Ayres) with no luck: ...
0
votes
1answer
62 views

$\nabla \times F=0$ implies that $F$ is conservative

Prove that if $F:\mathbb R^3\to \mathbb R^3$ is a vector field so that $\nabla\times F=0$ $\forall x\in \Omega\subset \mathbb R^3$ (where $\Omega$ is an open simply connected set), then $F$ is a ...
0
votes
1answer
20 views

Are Gradients of Radial Functions parallel?

For two radial functions $f(R)$ and $g(R)$ defined on an open set $U \subset \mathbb{R}^2$, is $\nabla f$ always parallel to $\nabla g$? Since both functions are constant around each circle $C_r(0)$ ...
0
votes
1answer
37 views

Region bounded by $x=y^2$ and $x=y^3$

What is meant by "the region bounded by $x=y^2$ and $x=y^3$"? The graphs of these two curves split the plane into 4 sections, but none of these are really bounded, they all kind of continue forever in ...
2
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0answers
24 views

Partial derivative of $F{\equiv}0$

If you have a function $F{\equiv}0$ then is the partial derivative of $F$ with respect to any of its variables $0$? Specifically, when we have Charpit's equations for a PDE $F(x,y,u,p,q) = 0$, where ...
1
vote
1answer
35 views

How can I go about solving this group of equations in as simple a way as possible?

They arise from partial derivatives of the Lagrange multiplier function. Here below is the original problem: Goal function: $$f(x,y,z)=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} $$ with ...
1
vote
1answer
54 views

Fundamental Theorem of Calculus With Function Containing Limit Variable

I'm trying to solve the following question: Evaluate $$\frac{\mathrm{d} }{\mathrm{d} s} \int^s_0 e^{st^2} dt $$ My thinking was that by the fundamental theorem of calculus, we have $ F(s) = ...
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0answers
24 views

Show that $\iint_S (n\times \nabla)f\, dS=\int_C tf\, ds$ and $\iint_S (n\times \nabla)\times f \,dS=\int_C t\times F\, ds$

If $S$ be a closed region lying on a surface and bounded by the curve $C$ and $n$ be the unit positive normal vector to $S$, and $f$ and $F$ be two fields with continuous fiest derivatives in $S$, ...
3
votes
2answers
52 views

Prove that $\int_S n\times r dS=0$

If $r$ be the position vector of a point on a closed surface $S$ and $n$ be the unit normal (outward) vector to $S$, then prove that $$\int_S n\times r\,dS=0$$ Attempt: $r=xi+yj+zk$, ...
0
votes
0answers
27 views

A problem of vector integration: Show that $\iint_S f grad f \times dS =0$

For any scalar field $f$, show that $\iint_S f\, \nabla f \times dS =0$. I don't have an idea to solve. Please help me.
2
votes
1answer
36 views

Is level set near a maximum value a circle?

Let $f$ be a $C^2$ function defined on $[0,1] \times [0,1]$. Let $0 \leq f(x) < 1$ on $[0,1] \times [0,1] \setminus \left(\frac{1}{2},\frac{1}{2}\right)$ and $f(\frac{1}{2},\frac{1}{2})=1$. It has ...
1
vote
1answer
31 views

Proving that the norm of $f'(y)$ is attained at $\pm\frac{\nabla f(y)}{\|\nabla f(y)\|}$.

Consider a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$ and a point $y\in \mathbb{R}^n$ such that $\nabla f(y)\neq 0$. Prove that there exists an unit vector $x_0\in\mathbb{R}^n$ such that ...
4
votes
3answers
112 views

Proof of this definite integral?

Saw this sometime in my calculus book, from the Putnam Math Challenges listed: $$\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1 }{ \underbrace{\dots}_{n-3 \, times} \int _{ 0 }^{ ...