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).
32,657
questions
3
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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/...
1
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1
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21
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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)=...
- 13
-1
votes
0
answers
10
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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 ...
- 11
0
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0
answers
10
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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
$$E(x_1,...
- 23.1k
0
votes
0
answers
15
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How to plot the vector field $\frac{1}{s}\hat{\phi}$ over the cylindrical coordinate system?
I am asked to sketch the vector field $\frac{1}{s}\hat{\phi}$ over the cylindrical coordinate system. However, I have no idea how to do it. Is there any relevant example?
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1
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0
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29
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Are these two triple integrals the same?
I need to write an iterated triple integral to represent the volume of the intersection of two balls in cartesian coordinates.
The two balls are $x^2+y^2+z^2\leq 9$ and $x^2+y^2+(z-8)^2 \leq 49$
I ...
- 602
1
vote
0
answers
17
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Finding which integrals are equal to the given triple integral
Which of the integrals are equal to
$$\int_{0}^{3}\int_{0}^{2}\int_{0}^{y} f(x,y,z) dz dy dx$$
a) $\int_{0}^{2}\int_{0}^{3}\int_{0}^{y} f(x,y,z) dz dy dx$
b) $\int_{0}^{3}\int_{0}^{2}\int_{0}^{y} f(x,...
- 726
6
votes
0
answers
55
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Using change of coordinates to find the exact value of an integral
Use an appropriate change of coordinates to find the exact value of the integral
$$\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$$
My work so far:...
- 1,468
2
votes
2
answers
68
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Integral of $\frac1{|x|^3}$ on a circular segment
How can I evaluate
$$\int \int _C |(x,y)|^{-3} dx dy$$
Where $C$ is the part between the chord $AB$ and the arc $AB$, and $|(x,y)| = \sqrt{x^2 + y^2}$? The radius of the circle is $R$.
I tried using ...
- 175
0
votes
0
answers
25
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How do we derive this formula?! (orthogonal complement involved)
I am studying differential geometry on manifolds at the moment and the following is a part of some notes.
Let $\psi:B^2(0,1) \to \mathcal S_+^2$, $\;w \mapsto (\sqrt{1-\vert w
\vert^2},w)$ be a ...
- 2,692
0
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1
answer
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Which of the following is the incorrect solution in finding the limit of $\frac{x^2}{x-y}$ as ${(x,y) \to (0,0)}$?
I have been solving the limits of functions of two variables to review for my upcoming test.
I have two solutions with different answers. I am trying to review both solutions, but it seems both have ...
- 467
1
vote
0
answers
18
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Triple integral of $\nabla \cdot \vec F$ over a cube
Evaluate $\iiint \nabla \cdot \vec F dV$
where $\vec F = 𝑥^2 \hat i + y^2 \hat j + z^2 \hat k$ and if $V$ is the
volume of the region enclosed by the cube $0≤ 𝑥 ≤ 1, 0≤ 𝑦 ≤
1,0≤ 𝑧 ≤ 1.$
2
votes
1
answer
46
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Unsure how to solve this integral
Let $$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.$$...
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0
votes
1
answer
40
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If $X \subset \mathbb{R}^n$ is compact then $X \times X \subset \mathbb{R}^n \times \mathbb{R}^n$ is compact - proof verification
Claim: If $X \subset \mathbb{R}^n$ is compact, then $X \times X \subset \mathbb{R}^n \times \mathbb{R}^n$ is compact.
Key definitions:
A set $X$ is closed if every convergence sequence converges to ...
- 5,345
0
votes
0
answers
6
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Orientation of Boundary Curves
In general, if we have some boundary curve D to a surface S where D=∂S and S has upwards pointing normal vectors, does this always imply positive (counter-clockwise) orientation? Is the opposite ...
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2
votes
2
answers
40
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What is the cartesian equivalent of $r=0$?
The question I have asks to convert
$$\int_{\pi/4}^{\pi/2} \int_{0}^{2/\sin{\theta}} r^{3/2} dr d\theta$$
I think I understand how to do most of it:
$r=\frac{2}{\sin{\theta}}\implies r\sin\theta=2 \...
- 1,468
2
votes
0
answers
34
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Determining which double integral has the largest area
Let $f(x,y)$ be a positive function. If the integrals
(A) $\int_{0}^{1} \int_{x^2}^{1} f(x,y) dydx$
(B) $\int_{0}^{1} \int_{x^3}^{1} f(x,y) dydx$
(C) $\int_{0}^{1} \int_{0}^{1} f(x,y) dydx$
are ranked ...
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3
votes
0
answers
32
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Converting a double integral from cartesian to polar coordinates
Question: The integral $$\int\limits_0^3 \int\limits_{|x|}^{\sqrt{18-x^2}} \sqrt{x^2+y^2} dy\ dx$$
has the following form in polar coordinates
My work so far:
$y=\sqrt{18-x^2} \implies x^2+y^2=18 \...
- 1,468
1
vote
0
answers
13
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Differential $1$-form and proof of an open disc and open circular annulus not being diffeomorphic
There is one example in my script about an application of a differential $1$ form in proving some subsets of $\Bbb R^2$ aren't diffeomorphic. As far as I've understood the explanation, we used a $C^2$ ...
0
votes
0
answers
19
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How does one find a parametrization of a curve , given the tangent vector and level set
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ be a $\mathcal{C}^1$ function, i.e the directional derivatives exists and are continuous.
Let $\vec{a}$ be a point in the non-empty regular level set $f=...
3
votes
0
answers
51
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True or False question for Fubini's Theorem
a) Is it true that $$\iint_{R} f(x,y)dydx=\iint_{R} f(x,y)dxdy$$
I thought this would be true because of Fubini's Theorem however Fubini's Theorem requires $f(x,y)$ to be continuous on $R$. There is ...
- 1,468
-1
votes
0
answers
51
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Continuity of $\theta(x,y) = \frac{(x,y)}{\sqrt{x^{2} + y^{2}}}$ in $\mathbb R^2 - \{ (0,0) \}$ using epsilon and delta definition
I recently tried to prove the continuity of this function $\theta(x,y) = \frac{(x,y)}{\sqrt{x^{2} + y^{2}}}$ in $\mathbb R^{2} - \{ (0,0) \}$ using the inverse image of some open in $\mathbb R^2$ but ...
2
votes
2
answers
44
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Abstract exercise about the Implicit Function Theorem
I am practicing Calculus problems and the following one caught my attention:
Let $F:\mathbb{R}^3\to \mathbb{R}$ a $\cal{C}^1$ function, and let $P\in \mathbb{R}^3$ such that $F(P)=0$, $\dfrac{\partial ...
- 175
0
votes
1
answer
23
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Computing the average value of a function with a double integral
Consider the integral $$\int_{0}^{1} \int_{\sqrt{y}}^{1} f(x,y) dxdy$$
Compute the average value of $f(x,y)=\sqrt{1+x^3}$ on the region R
The average value of a function is $\frac{1}{A}\iint f(x,y) ...
- 1,468
-1
votes
2
answers
65
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Show that the absolute maximum of $f(x, y) = \frac{(ax+by+c)^2}{x^2+y^2+1}$ is $a^2 + b^2 + c^2$
I got this question on my exam today: show that the function $f(x, y) = \dfrac{(ax+by+c)^2}{x^2+y^2+1}$ has an absolute maximum whose value is $a^2 + b^2 + c^2$.
I tried setting the gradient to the ...
- 21
4
votes
0
answers
25
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Multivariable function their variables depend on another one?
$if$ $z$$=$$F(x,y)$ , $y$$=$$g(t)$ and $x$$=$$h(t)$
Then we can say that the function $z$ is just a function of a single variable which is $t$ ??
$Because$ in this case the variables $x$ and $y$ are ...
1
vote
0
answers
36
views
Gradient of Frobenius inner product involving a homogeneous function
Let $f:\mathbb{R}^{m\times n} \rightarrow \mathbb{R}^{m \times n}$ be a differentiable function. I want to take the gradient of the Frobenius inner product between $X$ and $f(X)$ with respect to $X$. ...
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0
votes
0
answers
15
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Maximal open subset in which a function is injective
Consider the function $H(x,y) = (x^2-y^2, 2xy)$ with domain all the plane except the origin. I was asked to study its injectivity and local injectivity. It is easy to note that $H(x,y) = H(-x,-y)$, so ...
- 175
1
vote
1
answer
29
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estimate of measure of a $C^1$ range set
Let $\Omega\subset\mathbb{R}^n$ be an open set and
$f:\Omega\to\mathbb{R}^n$ be a $C^1$ function . Denote
$J_f(x)=\det(f'(x))$ . Prove that $$\mu(f(X))\leq\int_X|J_f(x)|dx$$
for any $X\subset\Omega$ ...
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1
vote
1
answer
26
views
Determinate $d^{k}f_{c}$ for function $f(x,y)$ with k and c given
In my Calculus 3 class I have been given following problem:
Determinate: $d^{k}f_{c}$ for function $f(x,y)$ where $c = (c1,c2)$ and $k = 2$.
My problem is that I have no clue what does $d^{k}f_{c}$ ...
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0
votes
1
answer
16
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Prove that the equation has solutions in two different intervals [-1,1] and [1,2]
$a,b,c \in \mathbb{R}$ and $a,b,c$ positive numbers
$\frac{(a+b)\cdot x+a-b}{x^2-1}+\frac{c}{x-2}=1$
Show that the equation has a solution in the interval $[-1,1]$ and a solution in the interval $[1,2]...
- 433
1
vote
1
answer
47
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How to express a function into powers of $(x-1)$ and $(y-2)$ using Taylor's formula?
Use Taylor's formula to express the following in powers of $(x-1)$ and $(y-2)$:
$f(x,y)=x^3 + y^3 + xy^2$
Solution:
$f(1,2)=1 +8 + 4=13$
$f_x (1,2) = 3 + 4=7$
$f_y (1,2) = 12 + 4=16$
$f_{xx} (1,2) = 6$...
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5
votes
1
answer
40
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What does a double integral represent? Please help me check my answers
I have three double integrals:
(A) $$\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$$
These need to be matched to the appropriate ...
- 1,468
1
vote
2
answers
36
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Surface Integrals for Calculating Volume
I understand that volume under a curve can be calculated with double integrals in Multivariable Calculus, but can it also be calculated with surface integrals? I would think that taking the surface ...
- 76
1
vote
0
answers
67
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Find the Lipschitz constant
The question:
Let $f(x) = ln(x + \sqrt{3 + x^2}) + \frac{1}{2}(x-2)^2$ with no restrictions on domain.
Determine the Lipschitz constant $L$ for the basic statement:
$$| \nabla f(x) - \nabla f(y)| \...
- 23
1
vote
0
answers
24
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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 ...
- 55
0
votes
0
answers
21
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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 $y, z \in \mathbb{R}^d$:
$$\int_x \exp(-f(x, y)) dx = \int_x \...
- 1
3
votes
2
answers
54
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Compute $\lim\limits_{(x,y) \to (0,0)} \frac{x^2}{x^2+y^2} \sin\big(\frac{xy}{\sqrt{x^2 + y^2}}\big)$
Acordding to Wolfranalpha:
$$\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}\;\; \...
- 605
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 ...
- 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:
$$I = \int_{a}^{b}dx_{1}\int_{a}^{x_{1}}...
- 397
7
votes
1
answer
33
views
Partial derivative of a function their variables depend on each other
if $z=F\left(x,y\right)$ and $y$=$\phi \left(x\right)$
Then is it correct to say that $z$ is just a function of a single variable which is $x$ ?
and if we try to compute $\frac{\partial z}{\...
- 379
1
vote
1
answer
32
views
Where is my mistake in showing $\nabla \mathbf{a}^{T}\mathbf{x}=a_{1}+a_{2}+\cdots+a_{n}$?
Let $f:=\mathbf{a}^{T}\mathbf{x}$. The claim that:
$$\nabla f=\nabla\mathbf{a}^{T}\mathbf{x}=\|\mathbf{a}\|_{1}=a_{1}+a_{2}+\cdots+a_{n}\tag{1}$$
is false where in fact the true answer is $\mathbf{a}^{...
- 343
0
votes
1
answer
64
views
How to transform triple integral $\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$
I have stumbled across this triple integral
$$\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}{...
- 19
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
17
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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}{...
- 7
0
votes
1
answer
18
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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 ...
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 $y=f(x)$, $$\frac{dy}{dx}=-\frac{\frac{\partial }{\partial x}\left(F\right)}{\frac{\partial }{\...
-4
votes
0
answers
42
views
How do I go about this first and second order partial derivatives question? [closed]
Problem
Let $f(x,y)$ be a function for which all its first and second order partial derivatives exist and are continuous and let ${\bf v}=\langle a,b\rangle$. Express $$(D_{\bf v}(D_{\bf v}f))(x,y),$$...
- 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 ...
- 47
1
vote
1
answer
25
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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 ...
- 23.1k