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).

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If $F$ and $f$ are continuous show that $g$ is continuous

Let $F:[a,b]\times [a,b] \rightarrow \mathbb{R}$ and $f:[a,b] \rightarrow \mathbb{R}$ be two continuous functions, let $g:[a,b] \rightarrow \mathbb{R}$ defined as: $$ g(x) = \int_a^b F(x,y) f(y) dy$$ ...
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Stokes theorem on surfaces

Im doing Stokes's theorem example I can't figure what book is saying. I get $curlF(x,y,z)=2i+k$. But why $ndA=(-yi-xj+k)$? enter image description here
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Differentiability of $f(x,y)=\frac{xy}{x^2+y^2}$ at $(1,1)$

I tried to show that $f(x,y)=\frac{xy}{x^2+y^2}$ is differentiable at $(1,1)$ by doing the following but I got stuck: $$\lim_{(h,k)\longrightarrow(0,0)}\frac{f(1+h,1+k)-f(1,1)-f_x(1,1)h-f_y(1,1)k}{\...
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For $F:R^{n} \to R^{m}, m>n$, with local inverse $G$ at $x$ such that $G(F(x)) = x$, is $DG$ a left inverse of $DF$? Why/Why not?

Assuming both functions are differentiable in some neighboorhood, under which conditions, if any, is the Jacobian of $G$ the left inverse of the Jacobian of $F$, ie $(JG)(JF) = Id^{nxn}$? Since we're ...
gaaaaaaaaaah's user avatar
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Dependence on a variable involving homogeneous functions of the 1st degree

Suppose that $f(x,z)$ and $g(x,y)$ are homogeneous rational functions of the 1st degree, that is, $f(tx,tz)=tf(x,z)$ and $g(tx,ty)=tf(x,y)$ and consider the function $$ F(x,y,z)=\big[f(x,z)+ay\big]\...
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Manifold with the same dimension as $\mathbb{R}^n$

I am studying manifolds at a basic level, and I was wondering if, when a set has the same dimension as the space where it is defined (take, for instance, $\{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 ...
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How to divide a catenary curve into parts of equal length?

I know the basic equation of a catenary is y = a*cosh((x-x0)/a)+b Length of a catenary curve is L = a*sinh((x-x0)/a) where x0 is a symmetry point or vertex or lowest x co-ordinate of a curve. I can ...
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What is the Divergence of a Spherically Symmetric Vector Fields?

A vector field is spherically symmetric about the origin if, on every sphere centered at the origin, it has constant magnitude and points either away from or toward the origin. A vector field that is ...
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Bounded second derivative also bounds the function

I've been struggling with this problem for a few days: Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined as $f(x,y)=xg(y)-yg(x)$, where $g:\mathbb{R}\rightarrow \mathbb{R}$ is such that $g\in C^2$, $...
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find the volume of the cylinder $x^2 + y^2 = 4$ outside of the cone $z = \sqrt{x^2 + y^2}$ using spherical coordinates

I setup the triple integral as follows: $$ \int _{\frac{\pi }{4}}^{\frac{\pi }{2}}\:\int _0^{\frac{\pi }{2}}\:\int _0^2\:p^2sin\phi dpd\theta d\phi \: $$ When I evaluate this I get the answer as: $$ \...
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The center of mass of a semiellipsoid

I am trying to find the center of mass of a semiellipsoid using cylindrical coordinates. $$ \frac{r^2}{a^2}+\frac{z^2}{b^2} \leq 1 $$ with $z < 0$ and density = 1. I know that the center of mass is ...
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How to calculate the surface area of the elliptical paraboloid given by $z=x^2 + 4y^2$

Also, $z$ should lie in the closed interval of $[0,4]$. I know the general method to this - to find the function $\langle x, y, z(x,y)\rangle$ differentiated with respect to $x$, then with respect to $...
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Differentiability & Partial Derivatives: Existence at a point Vs. Continuity at a point

I'm a bit confused about the multivariable differentiability criterion and what you're supposed to check. Let me see if I have this correct. The theorem is: If the partials EXIST AND ARE CONTINUOUS ...
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How to find work done by the vector field?

Suppose we have a vector field $\vec{F}(x,y,z) = (yz)\vec{i} + (xz)\vec{j} + (xy)\vec{k}$, and we are asked to find the work done by this vector field along a smooth curve $C$, from points $A(-1, 3, 9)...
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Show that if $f(t) = (\cos t, \sin t)$ then $f^{-1}$ is not continuous (Is my proof right?)

Let $S^1= \{(x,y) \in \mathbb{R}^2 : x^2+y^2=1\}$ and $f:[0, 2\pi) \rightarrow S^1$ defined as $f(t) = (\cos t, \sin t)$ Show that its inverse $f^{-1}: S^1 \rightarrow [0, 2\pi)$ is not continuous. ...
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Curve connecting the north pole of a sphere to another point on the sphere

Consider the following parameterization of a sphere of radius $R>0$ given as $\Gamma:[0,2\pi]\times [0,\pi]\to\mathbb{R}^3$ given by $(\theta,\phi)\mapsto (R\cos\theta\sin\phi,R\sin\theta\sin\phi, ...
mathematica's user avatar
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Proving that a function is depend upon a specific variable.

Suppose that $f_1(x,z),f_2(y)$, $g_1(x,y),g_2(z)$ are non-constant rational functions and let $$ F(x,y,z)=\big[\overbrace{f_1(x,z)+f_2(y)}^{f}\big]\big[\overbrace{g_1(x,y)+g_2(z)}^{g}\big]. $$ I want ...
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Calculate an integration on a sphere

Suppose $A \in R^{n \times n}$ is a symmetry matrix, I want to calculate the following thing: \begin{equation} \int_{\|x\|_2^2 = 1} x^TAx d\sigma(x) \end{equation} i.e. I want to calculate the ...
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What would be a good successor to Spivak's book Calculus? A book on multivariable calculus with roughly the same level of rigor.

I took calculus I, II, and III a long time ago, I barely remember anything, so I want to review it. Also I would like it if this time around I learn more about the why of things, since when I took ...
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When does the magnitude of the gradient equal the surface area of the $dxdy$ patch?

Given a surface $S$ in $\mathbb R^3$, what is the relationship between the gradient (when $S$ is defined as a level curve of function $F: \mathbb R^3 \to \mathbb R$) and surface area? I noticed such a ...
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Contour plotting for constrained optimization

I am trying to find the minimum surface area for a bottle that has a maximum volume of $50$. These are the functions: \begin{align} \text{Surface Area} &= 2.5\pi x + \pi(y + 1.25) \left( (y - 1....
Ibrahim Alzir's user avatar
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Define $f(0,0)$ so that $f'_x(0,0)$ exists

I am stuck with this problem and do not know how to proceed. I have been dealing with Discrete Math for years and forgot many things about Calculus. Consider the function $f:\mathbb{R}\setminus \{(0,...
Kristina Dedndreaj's user avatar
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When can you substitute in a differential equation?

I'm working on a differential equation, but I am not allowed to substitute itself back in at a later step, could someone tell me why? Here's my work. Consider: $$\frac{dx(t)}{dt} = W(x(t)) + Q(x(t),y(...
amongus's user avatar
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$f,g$ are real-valued, $f(x,y)=g(h_1(x),h_2(y))$, $f,h_i$ are continuous, $g$ is increasing, does $g$ must be continuous?

Assumptions: $f,g$ are real-valued. $g:[0,1]^2\to\mathbb R$. Functions $h_1:X\to\mathbb [0,1]$ and $h_2:Y\to\mathbb [0,1]$ are surjective continuous. $X,Y$ are connected separable. $f$ is continuous, $...
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How can I compute the $f^{-1}(q)$ (as a set) of an ellipsoid? Differential Geometry [closed]

How can I compute the $f^{-1}(q)$ (as a set) of an ellipsoid?
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Partial derivatives of functions with implicit variables

Consider a function $f(x,y)$ with $x$ being a function of $y$, that is $f(x(y), y)$. I would like to compute the gradient of this function $\nabla f = (\partial_x f, \partial_y f)$. Using the change ...
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Continuous and increasing in every variable does not imply continuous?

Let $f:\mathbb R^2\to \mathbb R$ $z=f(x,y)$ $f$ is continuous and strictly increasing in both $x,y$. It is known that continuous in each linear directions does not imply continuity. But the examples ...
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Set up intergal produced by force field $F(x,y)$ [closed]

Set up but do not evaluate the integral to compute the work produced by the force field $F(x,y) = \langle x+2y, x-y\rangle$ moving an object along the curve $y = \sqrt{x-2}$ traversed from $(3,1)$ to $...
Sara Ra's user avatar
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Work done by the force $\mathbf{F}$ to move a particle along the curve $\mathbf{r}$.

Consider the vector field $ \mathbf{F} =\Big(-\frac{y}{x^2+y^2}, \frac{x}{x^2+y^2}\Big) $ and the curve defined by $ \frac{x^2}{9} + \frac{y^2}{1} + z = 2 $. Calculate the work done by the force $ \...
user1259404's user avatar
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Are (0,y) saddle points of f(x,y)=yx|x|? [closed]

I have the function f(x,y) = yx|x| Is it correct that all the points of x=0 are saddle points for f?
applix's user avatar
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Does chain rule hold for closed rectangles as well?

Let $F:A\to \mathbb{R}$ be a function where $A = I_1\times \dots\times I_n\subseteq \mathbb{R}^n$ is a not necessarily open rectangle. Suppose that partial derivatives $F_i = \frac{\partial F}{\...
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2 votes
1 answer
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How to prove $\int_{\mathbb R} \nabla_u F (x, u_k(x))\cdot\varphi dx \to \int_{\mathbb R} \nabla_u F (x, u(x))\cdot\varphi dx$ for all test function?

Let $F(x, u)\in C^1(\mathbb R\times\mathbb R^n, \mathbb R)$ and let $(u_k)$ be a sequence such that $$ u_k\to u \text{ in } L^\infty_{loc}(\mathbb R, \mathbb R^n)$$ and $$ u_k\to u \text{ a.e. in } \...
Physics user's user avatar
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Applying the fundamental theorem of calculus in double integral

How would one go about calculating $$\frac{d}{dt}\int^t_{-t}f(z,t)dt.$$ And more specifically, $$\frac{d}{dt}\int^t_{-t}\int^t_{-t}f(x,y)dxdy$$ Assuming the necessary conditions, i got to $$lim_{h\to ...
user670565's user avatar
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Evaluate the following integrals:

Evaluate the integrals: I =$\iiint(2x-y+z^2)dV$ where V is the ellipsoid $3x^2 +4y^2 + 5z^2 = 17.$ I tried converting the equation of the ellipsoid to Spherical Coordinates and got to 3$\rho^2cos^2\...
MathDude446's user avatar
2 votes
1 answer
43 views

Convergence of $\int_0^1 \int_0^1 (1-xy)^{-a}dxdy$

How can I investigate the convergence of $\int_0^1 \int_0^1 (1-xy)^{-a}dxdy$ where $a \in (0, \infty)$? I know that one approach is to explicitly find the antiderivative but that seemed too laborious. ...
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When $f/\frac{\partial f}{\partial x}$ does not depend upon $x$?

Suppose that $f(x,y,\ldots)$ is a multivariable non-constant rational function. When $f/\frac{\partial f}{\partial x}$ does not depend upon $x$? My attempt: The question equivalent to when $$ \frac{\...
boaz's user avatar
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2 votes
2 answers
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Prove $\frac{1}{\sqrt{a+b+7c}}+\frac{1}{\sqrt{c+b+7a}}+\frac{1}{\sqrt{a+c+7b}}\ge 1.$

Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c+abc=4.$ Prove that$$\color{black}{\frac{1}{\sqrt{a+b+7c}}+\frac{1}{\sqrt{c+b+7a}}+\frac{1}{\sqrt{a+c+7b}}\ge 1.}$$ I found the ...
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1 vote
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Einstein notation and differential operators.

So I am dealing with the differential operator $\mathbf{D} = \mathbf{r} \times \boldsymbol{\nabla}$ where $r = x_i \mathbf{e}_i$. We then introduce $(\mathbf{D}f)_i$, which can be expressed as $\...
user1250010's user avatar
1 vote
1 answer
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A question about the dependence of a function on a variable.

Suppose that $f_1(x),f_2(x)$, $g_1(y),g_2(y)$ are non-constant rational functions and let $$ F(x,y)=(f_1(x)+g_1(y))(f_2(x)+g_2(y)). $$ I want to prove that $F$ must depend upon $x$, that is, that $\...
boaz's user avatar
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Is Maxwell's function for describing the magnetic field around a current an exception to the rule that exact differentials are path independent?

In section 480 of his Treatise on Electricity and Magnetism Maxwell argues that the differential describing the work per pole strength around a current in an infinite wire is the differential of the ...
SAG's user avatar
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Finding the centroid of a solid region using triple integrals

I have been struggling with this textbook problem for some time now and thought I'd seek assistance. The question asks me to find the centroid of a solid region bounded by these two equations: $$z = \...
pyrate7c's user avatar
2 votes
0 answers
40 views

Verify (or critique) this informal proof of Green's theorem

In order to better understand Green's Theorem, I developed this informal proof, which I request verification and critique of (both the proof and its writing). Of course, any textbook has a proof: my ...
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find the surface area of the portions of the sphere $x^2+y^2+z^2 = a^2$ within the cylinder $x^2+y^2=ay$ [closed]

So I have gotten this far: $$ \int_0^{\pi} \int_0^{a\sin\theta } \!\! \frac{a}{\sqrt{a^2-r^2}}r \, dr \, d\theta $$ In the other solutions I see online for the $d\theta$ integral people evaluate it ...
Mattel's user avatar
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What is the change in angle of of a symmetric matrix when summed over a plane?

A symmetric matrix has orthogonal eigenvectors with real eigenvalues, and hence can be thought of as scaling along a particular orthogonal set of axes. Of course, not all vectors are (usually) ...
SRobertJames's user avatar
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Prove $f$ is differentiable at $M_n(\mathbb{R})$

Denote: $f:M_n(\mathbb{R}) \to M_n(\mathbb{R})$ such that $f(X)=X^3$. Prove $f$ is differentiable at $M_n(\mathbb{R})$ and find $(Df(A))(X) \quad \forall A,X \in M_n(\mathbb{R})$. My solution: ...
Algo's user avatar
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Change the order of integration: $\int_0^1 \int_0^1 \int_{x^2}^1 12 xz \exp(z y^2) \mathrm dy \mathrm dx \mathrm dz$

Consider the following integral: \begin{equation} \int_0^1 \int_0^1 \int_{x^2}^1 12 xz \exp(z y^2) \ \mathrm dy \ \mathrm dx \ \mathrm dz. \end{equation} It has been said that the integral cannot be ...
Mike Gotier's user avatar
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0 answers
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If $\Delta_3 u>0$ on $D$, prove $\max_{x\in D}u(x)=u(x_0)$ for some $x_0\in \partial D$

This is a problem in a PDEs course I am helping a student with, and I would like to take this opportunity to verify my solution. I will also entertain alternative/easier approaches, because I suspect ...
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2 votes
2 answers
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Is a vector field irrotational iff its Jacobian is symmetric?

Let $F: \mathbb R^n \to \mathbb R^n$ be a smooth vector field. I conjecture that $F$ is irrotational iff its Jacobian is a symmetric matrix. In two and three dimensions, this seems clear from Green ...
SRobertJames's user avatar
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1 vote
1 answer
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Question about the area element of picking points uniformly on a 3 dimensional sphere

Given the unit sphere in $\mathbb{R}^3$, we want to pick points uniformly across its surface. My first thought was to parameterize by spherical coordinates $$ (\theta,\phi) \mapsto (\cos\theta\sin \...
rosemary 2.0's user avatar
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0 answers
68 views

I may be like 'Don Quixote', but I want to do this integral. [closed]

The following φ is a parameter indication of the Mobius strip. Since Mobius strips cannot be oriented, the calculation of surface area is meaningless. As a matter of fact, I am smarter than Don ...
Blue Various's user avatar

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