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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is transposed vector times transposed vector possible?

I was wondering if it was indeed possible to perform a transposed vector multiplication with another transposed vector. And if so how I'm supposed to do so. Background: From https://en.wikipedia.org/...
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1 vote
1 answer
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Determine a linear approximation of $f$ where $z=f(x,y)$ and $e^{z} = xyz$ at $(e^{2}, 1/2)$

Let $z = f(x, y)$ be the function clearly defined by the equation $e^{z} = xyz$, determine the linear approximation of $z$ around the point $(e^2,1/2)$ What I know: I know I can use $L\left(x,y\right)=...
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-1 votes
0 answers
10 views

Matrix on Matrix deriviatives - transpose operations?

I was given the following exercise which I am not sure how to solve. Let there be matrices ${\bf X} \in \Bbb R^{64 \times 1024}$ and ${\bf W} \in \Bbb R^{512 \times 1024}$. Let ${\bf Y} = {\bf X} {\bf ...
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Are square configurations the only critical configurations of the energy on the circle?

$\newcommand{\S}{\mathbb{S}^1}$Let$$M=\{(x_1,x_2,x_3,x_4) \in \S\times \S \times \S \times \S\, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$ Let $E:M \to \mathbb{R}$ be defined by $$E(x_1,...
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How to plot the vector field $\frac{1}{s}\hat{\phi}$ over the cylindrical coordinate system?

I am asked to sketch the vector field $\frac{1}{s}\hat{\phi}$ over the cylindrical coordinate system. However, I have no idea how to do it. Is there any relevant example?
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1 vote
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29 views

Are these two triple integrals the same?

I need to write an iterated triple integral to represent the volume of the intersection of two balls in cartesian coordinates. The two balls are $x^2+y^2+z^2\leq 9$ and $x^2+y^2+(z-8)^2 \leq 49$ I ...
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1 vote
0 answers
17 views

Finding which integrals are equal to the given triple integral

Which of the integrals are equal to $$\int_{0}^{3}\int_{0}^{2}\int_{0}^{y} f(x,y,z) dz dy dx$$ a) $\int_{0}^{2}\int_{0}^{3}\int_{0}^{y} f(x,y,z) dz dy dx$ b) $\int_{0}^{3}\int_{0}^{2}\int_{0}^{y} f(x,...
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6 votes
0 answers
55 views

Using change of coordinates to find the exact value of an integral

Use an appropriate change of coordinates to find the exact value of the integral $$\int_{-\sqrt{3}}^{\sqrt{3}}\int_{-\sqrt{3-x^2}}^{\sqrt{3-x^2}}\int_{-3+x^2+y^2}^{3-x^2-y^2}x^2dzdydx$$ My work so far:...
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2 votes
2 answers
68 views

Integral of $\frac1{|x|^3}$ on a circular segment

How can I evaluate $$\int \int _C |(x,y)|^{-3} dx dy$$ Where $C$ is the part between the chord $AB$ and the arc $AB$, and $|(x,y)| = \sqrt{x^2 + y^2}$? The radius of the circle is $R$. I tried using ...
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0 votes
0 answers
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How do we derive this formula?! (orthogonal complement involved)

I am studying differential geometry on manifolds at the moment and the following is a part of some notes. Let $\psi:B^2(0,1) \to \mathcal S_+^2$, $\;w \mapsto (\sqrt{1-\vert w \vert^2},w)$ be a ...
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0 votes
1 answer
15 views

Which of the following is the incorrect solution in finding the limit of $\frac{x^2}{x-y}$ as ${(x,y) \to (0,0)}$?

I have been solving the limits of functions of two variables to review for my upcoming test. I have two solutions with different answers. I am trying to review both solutions, but it seems both have ...
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1 vote
0 answers
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Triple integral of $\nabla \cdot \vec F$ over a cube

Evaluate $\iiint \nabla \cdot \vec F dV$ where $\vec F = 𝑥^2 \hat i + y^2 \hat j + z^2 \hat k$ and if $V$ is the volume of the region enclosed by the cube $0≤ 𝑥 ≤ 1, 0≤ 𝑦 ≤ 1,0≤ 𝑧 ≤ 1.$
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2 votes
1 answer
46 views

Unsure how to solve this integral

Let $$F\left(x,y\right)=xi+e^{y^2}j.$$ Evaluate $$\int _CF\cdot dr,$$ where $C$ is the curve $$r\left(t\right)=\frac{1}{t^2+1}i+\left(3+10t^2\cos\left(\frac{\pi t}{2}\right)\right)j,\quad0\le t\le 1.$$...
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0 votes
1 answer
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If $X \subset \mathbb{R}^n$ is compact then $X \times X \subset \mathbb{R}^n \times \mathbb{R}^n$ is compact - proof verification

Claim: If $X \subset \mathbb{R}^n$ is compact, then $X \times X \subset \mathbb{R}^n \times \mathbb{R}^n$ is compact. Key definitions: A set $X$ is closed if every convergence sequence converges to ...
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0 votes
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Orientation of Boundary Curves

In general, if we have some boundary curve D to a surface S where D=∂S and S has upwards pointing normal vectors, does this always imply positive (counter-clockwise) orientation? Is the opposite ...
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2 votes
2 answers
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What is the cartesian equivalent of $r=0$?

The question I have asks to convert $$\int_{\pi/4}^{\pi/2} \int_{0}^{2/\sin{\theta}} r^{3/2} dr d\theta$$ I think I understand how to do most of it: $r=\frac{2}{\sin{\theta}}\implies r\sin\theta=2 \...
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2 votes
0 answers
34 views

Determining which double integral has the largest area

Let $f(x,y)$ be a positive function. If the integrals (A) $\int_{0}^{1} \int_{x^2}^{1} f(x,y) dydx$ (B) $\int_{0}^{1} \int_{x^3}^{1} f(x,y) dydx$ (C) $\int_{0}^{1} \int_{0}^{1} f(x,y) dydx$ are ranked ...
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3 votes
0 answers
32 views

Converting a double integral from cartesian to polar coordinates

Question: The integral $$\int\limits_0^3 \int\limits_{|x|}^{\sqrt{18-x^2}} \sqrt{x^2+y^2} dy\ dx$$ has the following form in polar coordinates My work so far: $y=\sqrt{18-x^2} \implies x^2+y^2=18 \...
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1 vote
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Differential $1$-form and proof of an open disc and open circular annulus not being diffeomorphic

There is one example in my script about an application of a differential $1$ form in proving some subsets of $\Bbb R^2$ aren't diffeomorphic. As far as I've understood the explanation, we used a $C^2$ ...
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0 answers
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How does one find a parametrization of a curve , given the tangent vector and level set

Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ be a $\mathcal{C}^1$ function, i.e the directional derivatives exists and are continuous. Let $\vec{a}$ be a point in the non-empty regular level set $f=...
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3 votes
0 answers
51 views

True or False question for Fubini's Theorem

a) Is it true that $$\iint_{R} f(x,y)dydx=\iint_{R} f(x,y)dxdy$$ I thought this would be true because of Fubini's Theorem however Fubini's Theorem requires $f(x,y)$ to be continuous on $R$. There is ...
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-1 votes
0 answers
51 views

Continuity of $\theta(x,y) = \frac{(x,y)}{\sqrt{x^{2} + y^{2}}}$ in $\mathbb R^2 - \{ (0,0) \}$ using epsilon and delta definition

I recently tried to prove the continuity of this function $\theta(x,y) = \frac{(x,y)}{\sqrt{x^{2} + y^{2}}}$ in $\mathbb R^{2} - \{ (0,0) \}$ using the inverse image of some open in $\mathbb R^2$ but ...
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2 votes
2 answers
44 views

Abstract exercise about the Implicit Function Theorem

I am practicing Calculus problems and the following one caught my attention: Let $F:\mathbb{R}^3\to \mathbb{R}$ a $\cal{C}^1$ function, and let $P\in \mathbb{R}^3$ such that $F(P)=0$, $\dfrac{\partial ...
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  • 175
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1 answer
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Computing the average value of a function with a double integral

Consider the integral $$\int_{0}^{1} \int_{\sqrt{y}}^{1} f(x,y) dxdy$$ Compute the average value of $f(x,y)=\sqrt{1+x^3}$ on the region R The average value of a function is $\frac{1}{A}\iint f(x,y) ...
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  • 1,468
-1 votes
2 answers
65 views

Show that the absolute maximum of $f(x, y) = \frac{(ax+by+c)^2}{x^2+y^2+1}$ is $a^2 + b^2 + c^2$

I got this question on my exam today: show that the function $f(x, y) = \dfrac{(ax+by+c)^2}{x^2+y^2+1}$ has an absolute maximum whose value is $a^2 + b^2 + c^2$. I tried setting the gradient to the ...
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4 votes
0 answers
25 views

Multivariable function their variables depend on another one?

$if$ $z$$=$$F(x,y)$ , $y$$=$$g(t)$ and $x$$=$$h(t)$ Then we can say that the function $z$ is just a function of a single variable which is $t$ ?? $Because$ in this case the variables $x$ and $y$ are ...
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1 vote
0 answers
36 views

Gradient of Frobenius inner product involving a homogeneous function

Let $f:\mathbb{R}^{m\times n} \rightarrow \mathbb{R}^{m \times n}$ be a differentiable function. I want to take the gradient of the Frobenius inner product between $X$ and $f(X)$ with respect to $X$. ...
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  • 55
0 votes
0 answers
15 views

Maximal open subset in which a function is injective

Consider the function $H(x,y) = (x^2-y^2, 2xy)$ with domain all the plane except the origin. I was asked to study its injectivity and local injectivity. It is easy to note that $H(x,y) = H(-x,-y)$, so ...
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  • 175
1 vote
1 answer
29 views

estimate of measure of a $C^1$ range set

Let $\Omega\subset\mathbb{R}^n$ be an open set and $f:\Omega\to\mathbb{R}^n$ be a $C^1$ function . Denote $J_f(x)=\det(f'(x))$ . Prove that $$\mu(f(X))\leq\int_X|J_f(x)|dx$$ for any $X\subset\Omega$ ...
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1 vote
1 answer
26 views

Determinate $d^{k}f_{c}$ for function $f(x,y)$ with k and c given

In my Calculus 3 class I have been given following problem: Determinate: $d^{k}f_{c}$ for function $f(x,y)$ where $c = (c1,c2)$ and $k = 2$. My problem is that I have no clue what does $d^{k}f_{c}$ ...
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0 votes
1 answer
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Prove that the equation has solutions in two different intervals [-1,1] and [1,2]

$a,b,c \in \mathbb{R}$ and $a,b,c$ positive numbers $\frac{(a+b)\cdot x+a-b}{x^2-1}+\frac{c}{x-2}=1$ Show that the equation has a solution in the interval $[-1,1]$ and a solution in the interval $[1,2]...
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  • 433
1 vote
1 answer
47 views

How to express a function into powers of $(x-1)$ and $(y-2)$ using Taylor's formula?

Use Taylor's formula to express the following in powers of $(x-1)$ and $(y-2)$: $f(x,y)=x^3 + y^3 + xy^2$ Solution: $f(1,2)=1 +8 + 4=13$ $f_x (1,2) = 3 + 4=7$ $f_y (1,2) = 12 + 4=16$ $f_{xx} (1,2) = 6$...
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  • 131
5 votes
1 answer
40 views

What does a double integral represent? Please help me check my answers

I have three double integrals: (A) $$\int_{0}^{1}\int_{x^2}^1 dydx$$ (B) $$\int_{0}^{1}\int_{x}^{2x} x^2 dydx$$ (C) $$\int_{0}^{1}\int_{-y}^{y} dxdy$$ These need to be matched to the appropriate ...
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1 vote
2 answers
36 views

Surface Integrals for Calculating Volume

I understand that volume under a curve can be calculated with double integrals in Multivariable Calculus, but can it also be calculated with surface integrals? I would think that taking the surface ...
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1 vote
0 answers
67 views

Find the Lipschitz constant

The question: Let $f(x) = ln(x + \sqrt{3 + x^2}) + \frac{1}{2}(x-2)^2$ with no restrictions on domain. Determine the Lipschitz constant $L$ for the basic statement: $$| \nabla f(x) - \nabla f(y)| \...
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1 vote
0 answers
24 views

Flux through surface of revolution

I'm trying to solve the following problem Let $C$ be the curve in the $xy$ plane given in polar coordinates by $r = 2-\sin(\theta),\ 0 \leq \theta \leq \pi$ and let $S$ be the surface given by ...
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  • 55
0 votes
0 answers
21 views

Integral preserving multivariable functions

I am trying to find sufficient conditions on functions $f: \mathbb{R}^d \times \mathbb{R}^d \longrightarrow \mathbb{R}$ such that for all $y, z \in \mathbb{R}^d$: $$\int_x \exp(-f(x, y)) dx = \int_x \...
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  • 1
3 votes
2 answers
54 views

Compute $\lim\limits_{(x,y) \to (0,0)} \frac{x^2}{x^2+y^2} \sin\big(\frac{xy}{\sqrt{x^2 + y^2}}\big)$

Acordding to Wolfranalpha: $$\lim\limits_{(x,y) \to (0,0)} \frac{x^2}{x^2+y^2} \sin\left(\frac{xy}{\sqrt{x^2 + y^2}}\right) \quad\text{does not exist.}$$ Using: $$\gamma(t) = (t,0)\;\; \text{and}\;\; \...
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0 votes
0 answers
17 views

How to know which double integral corresponds to which graph

This is the question This is what I did to sketch the regions of integration for (A), (B), and (C) So, for (a) The area of the triangle in the xy-plane corresponds to C (b) The area of a region in ...
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  • 1,468
1 vote
0 answers
30 views

Multiple integrals identity

I am trying to solve a problem from my homework and I am one step away of getting it done. Let $ a< b$ two real numbers and consider the multiple integral: $$I = \int_{a}^{b}dx_{1}\int_{a}^{x_{1}}...
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7 votes
1 answer
33 views

Partial derivative of a function their variables depend on each other

if $z=F\left(x,y\right)$ and $y$=$\phi \left(x\right)$ Then is it correct to say that $z$ is just a function of a single variable which is $x$ ? and if we try to compute $\frac{\partial z}{\...
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1 vote
1 answer
32 views

Where is my mistake in showing $\nabla \mathbf{a}^{T}\mathbf{x}=a_{1}+a_{2}+\cdots+a_{n}$?

Let $f:=\mathbf{a}^{T}\mathbf{x}$. The claim that: $$\nabla f=\nabla\mathbf{a}^{T}\mathbf{x}=\|\mathbf{a}\|_{1}=a_{1}+a_{2}+\cdots+a_{n}\tag{1}$$ is false where in fact the true answer is $\mathbf{a}^{...
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  • 343
0 votes
1 answer
64 views

How to transform triple integral $\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$

I have stumbled across this triple integral $$\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$$ where $$\Omega =\left\{(x,y,z)\in{\cal{R}}^3\ \bigg| \ \frac{x^2}{...
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0 votes
0 answers
8 views

higher order derivatives in NLS

While reading a text in multivariable calculus , a following definition of higher order derivatives in normed linear spaces is given like this : If $U\subset E$ is open in some normed linear space $E$...
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-1 votes
0 answers
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Determine the linear approximation of z about the point $(e^{2}, \frac{1}{2})$

Let $z = f(x,y)$ be the function implicitly defined by the equation $e^{z} = xyz$ and the solution $(e^{2}, \frac{1}{2}, 2)$ , determine the linear approximation of z about the point $(e^{2}, \frac{1}{...
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  • 7
0 votes
1 answer
18 views

Flux of a vector field across the upper unit hemisphere

I want to compute the flux of the vector field $F(x,y,z)=(y,x,x^3)$ across the hemisphere $S\ldots x^2+y^2+z^2=1.$ My thoughts: Since the orientation isn't mentioned, I took it to mean the unit ...
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1 vote
1 answer
40 views

implicit function theorem in multivariable function

My question that in multivariable calculus the implicit function theorem states that: if $F(x,y)$ and $y=f(x)$, $$\frac{dy}{dx}=-\frac{\frac{\partial }{\partial x}\left(F\right)}{\frac{\partial }{\...
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-4 votes
0 answers
42 views

How do I go about this first and second order partial derivatives question? [closed]

Problem Let $f(x,y)$ be a function for which all its first and second order partial derivatives exist and are continuous and let ${\bf v}=\langle a,b\rangle$. Express $$(D_{\bf v}(D_{\bf v}f))(x,y),$$...
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  • 1
0 votes
1 answer
48 views

Show the identity

Let $f:\mathbb{R^2}\rightarrow\mathbb{C}$ be a differentiable function. The function $F:\mathbb{C}\rightarrow\mathbb{C}$, $F(x+iy):=f(x,y)$ is given and the "partial derivative" operator is ...
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  • 47
1 vote
1 answer
25 views

A square on the equator of a sphere is a critical point of the electrostatic potential

$\newcommand{\S}{\mathbb{S}^2}$ This is a self-answered question. I learned something from spelling out the details, and I hope this could be interesting to others. I would welcome alternative ...
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