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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1answer
18 views

Addition of integrals with different variables

I came across to an interesting problem recently, which can be solved if it is assumed that I can add $N$ integrals defined on the same domain, but using different variables, together, under the same ...
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1answer
12 views

Gradient of squared distance to a convex set

I have the following problem: Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$, $f(x)=(\operatorname{dist}(x,D))^2$ where $D$ is a convex, close set in $\mathbb{R}^n$. Prove that $f$ is convex and ...
1
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0answers
63 views

eigenvalues of transformed Hessian

Let us define the vector $\mathbf y$ by $y_i := \exp(x_i)$, with $\mathbf x = (x_i)\in \mathbb{R}^N$, and $f : \mathbb{R}^N \rightarrow \mathbb{R}$, $$\displaystyle f\left(\mathbf x(\mathbf y)\right) ...
0
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1answer
20 views

Differentiating functions of multiples of dependant variables

Let $f \left( \frac{a(x)}{b(x)} \right)$ then what is $\frac{df}{dx}$ and $\frac{\partial f}{ \partial x}$? How did you come to this answer? I'm guessing it's some kind of combination between the ...
2
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0answers
24 views

What vector field property means “is the curl of another vector field?”

I know that a vector field $\mathbf{F}$ is called irrotational if $\nabla \times \mathbf{F} = \mathbf{0}$ and conservative if there exists a function $g$ such that $\nabla g = \mathbf{F}$. Under ...
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0answers
23 views

Find the surface area using double integral

Find the surface of $z^2=2x$, which lies within the cylinder $y^2+(2x-0.5)^2=0.5^2$. By using $ \iint_{R} \sqrt{1+z_x^2+z_y^2}dxdy=\iint_{R} \sqrt{1+\frac{1}{2x}}dxdy $ $ z^2=2x \Rightarrow ...
0
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1answer
28 views

Verify Green’s Theorem for the vector field F = x i + y j and the region Ω

Verify Green’s Theorem for the vector field F = x i + y j and the region Ω being the part below the diagonal y = 1 − x of the unit square with the lower left corner at the origin. i) Sketch the ...
1
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1answer
28 views

Evaluating double integrals by inspection

I'm trying to use as much information about the domain to be able to solve the integral without actually integrating. The problem is: Evaluate $\oint\limits_C {(x\sin ({y^2}) - {y^2})dx + ...
1
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0answers
12 views

Defining a multiple integral on non-rectangular regions

Usually the Riemann integral for $\mathbb{R^n}$ is defined on a hyperrectangular region $T$, by partitioning the region's "edges", which are $(a_1,b_1) \times (a_2,b_2) \times \dots \times (a_n,b_n)$ ...
4
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4answers
92 views

Evaluate $\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$ using a double integral

I was given the following problem: Evaluate the following integrate using a double integral: $\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$. The professor told us off the bat the answer was ...
0
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1answer
28 views

mean distance from a point in a circle to its boundary ( circumference)?

I have perused the solution to the average distance from a point in a ball to a point on its boundary. I don't quite understand it. However, it seems likely that the analogous problem 'Average ...
1
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0answers
27 views

volume involving cross section

Find the volume of the region with base enclosed by $y = x^2$ and $y = 3$ and cross sections perpendicular to the $y$-axis are rectangles of height $y^2$. I sketched the graph and set-up the ...
0
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1answer
19 views

proof that this is of class $C^{+\infty}$

consider $f(x,y,z)=x^4+2x\cos y+\sin z$, proof that in a neighborhood of $0$, the equation $f(x,y,z)=0$ sets $z$ as a function of class $C^{\infty}$ of the variables $x,y$. compute $\frac{\partial ...
1
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0answers
13 views

Determining the Domain and Range of a multi-dimensional function

$ f(x,y,z) = \frac{3}{2-x} + \frac{1}{y-z} $ i) Write down the domain of $f$ ii) Determine the range $T$ of $f$. For each $c \in T$ find a point $(x,y,z) \in \mathbb{R}^3$ such that $f(x,y,z) = 1 $. ...
1
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1answer
32 views

How to find gradient in other coordinate systems?

I forgot the following thing and I don't seem to find it anywhere on the internet Let $u=f(x,y)$ and $v=g(x,y)$. What is the gradient of $F(u,v)$? Thanks in advance.
1
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1answer
20 views

Help with finding the associated potential function

I have the following vector field $$ F = (z^3+x^3,2y,3z^2x)$$ I need to calculate the work from the origin (0,0,0) to the point (2,-1,-2) I'm not sure if it's correct but I found the associated ...
1
vote
1answer
21 views

Finding the intersection of a line and hyperplane

Taking the hyperplane P = $\{ $x$ :3x^1-3x^2-3x^3-x^4=0\}$ and the line $t(e_1-e_2)+(1-t)e_4$ I do not know how to solve for t, where t is the intersection of the two. This is problem 2b from ...
0
votes
1answer
27 views

Find the work done by $F$ when it moves a particle from the origin to the point (2,-1,-2) on $C$, by performing a line integra

I have the following question and I am not 100% sure on how to proceed. I think I have answered a) but I'm not entirely sure if it's correct Consider the path $C$ which follows the curve ...
0
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1answer
25 views

computing the area of an ellipse $x^2-2xy+2y^2+4y=8$ using a specified change of variables

I am trying to calculate the area of an enclosed ellipse $x^2-2xy+2y^2+4y=8$ as an integral in the variables $u=x-y$ and $v=y+2$. To be completely honest I am at a loss for ideas here I am not sure ...
1
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1answer
38 views

Evaluating the line integral $\oint\limits_C \ {x-y\over x^2+y^2}dx+{x+y\over x^2+y^2}dy$ over a closed path $C$

Let $C$ be the closed path formed by the line $x=2$ and the parabola $y^2=2(x+2)$. Find $$\oint\limits_C \ {x-y\over x^2+y^2}dx+{x+y\over x^2+y^2}dy.$$ First I tried to find this integral for the ...
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2answers
24 views

Double Integral of $(x^2+y^2)^{-3/2}$ over a particular domain

I am trying to calculate the double integral of $\displaystyle \int\int_{\mathcal{D}}(x^2+y^2)^{-3/2}dxdy$ where the domain $\mathcal{D} =\{(x,y):x^2+y^2\leq1, x+y\geq 1\}$. I know that polar ...
3
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2answers
48 views

Proving $f$ is continuous using the $\epsilon-\delta$ definition

Prove that $f(x,y) = x^2 + xy$ is continuous using the $\epsilon-\delta$ definition, at $(1,-1).$ I encountered this question in a test and I fumbled through it quite shamelessly. Here is my latest ...
2
votes
2answers
33 views

Verify that the set $\Omega = \lbrace (u,v) \in \mathbb{R}^2 \mid |u| + |v| \leq 1 \rbrace$ is Jordan measurable

Motivation: I am currently in a rather uncomfortable spot in my Analysis studies. In class we introduced the Jordan measure in a very vague way, meaning no proofs, no examples. (Because next Semester ...
0
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2answers
27 views

Why does this relationship hold (integrals)

If I define $$P(t,T) = \exp\{ \int _t^Tf(t,s)ds\}$$ then why is it true that $$\ln P(0,T) - \ln P(0,t) = - \int_t^T f(0,s)ds \tag{1}$$ We would have $$P(0,T) = \exp \{ - \int_0^T f(0,s)ds\}$$ ...
0
votes
1answer
16 views

Integration over a combination of sphere and cone

If $P$ is the part of the sphere $x^2+y^2+z^2=4$ that is above the cone $z=\sqrt{x^2+y^2}$, them what is $$\int_P yz \,dS?$$ I can't really visualize this problem and I'm not sure how to ...
2
votes
1answer
27 views

The Leibniz rule for the curl of the product of a scalar field and a vector field

I have some scalar field $u:D \rightarrow\mathbb R; \space \space D\subset \mathbb R^3$ and a vector field $\vec{v}: D\rightarrow \mathbb R^3$ and I want to show that: ...
-1
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0answers
33 views

Find an example of critical point

Find an example have the following property: Let $\Omega $ be open in $\mathbb{R}^{n}$, $f, g : \Omega \rightarrow \mathbb {R}$ be $\mathcal{C}^{1}(\Omega)$ and $S=\begin{Bmatrix} ...
0
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0answers
24 views

Find the flux of $F=m\vec r/|\vec r|^3$ out of the surface of the cube

The problem I have trouble with is: Find the flux of $F=m\vec r/|\vec r|^3$ out of the surface of the cube $-a\le x, y, z\le a$ The answer is $4\pi m$, which does not make sense to me. I guess ...
5
votes
3answers
72 views

Limit of $\frac{\sin(x+y)}{x+y}$ as (x,y)→(0,0)

$$ \lim\limits_{(x, y)\to (0, 0)}\frac{\sin(x+y)}{x+y} $$ I did the following $a)$ along $x$ axis, the limit is one $b)$ along $y$ axis the limit is one $c)$ along $y=x$ the limit is one Since ...
-1
votes
1answer
23 views

What is the best fit (in the sense of least-squares) to the data?

A) Find the best fit (in the sense of least-squares) to the data $x_1$ $(1,-1,-1,1)$ $x_2$ $(1,1,-1,-1)$ $y$ $(5,1,1,1)$ by a linear function of the form $y$=$a$+$bx_1$+$cx_2$ B) Find ...
1
vote
1answer
42 views

Let $V$ be a subspace of $R^3$ spanned by the vectors $(1,2,1)$ and $(2,1,2)$

a) Find an orthogonal basis for $V$. b) Find the projection of the vector $(1,3,0)$ onto $V$. c) Find the distance of the vector $(1,3,0)$ from $V$. Alright, i think i got it, but i guess what I'm ...
2
votes
2answers
32 views

Suppose $\{v_1,v_2,v_3\}$ is a basis for some subspace $V$ of $\mathbb R^m$.

Let $b$ be a vector in that subspace. Prove that if $b$ is orthogonal to all three basis vectors, then b has to be a zero vector. Hint: What is $\|b\|$ I do not know how to start this proof. Thanks ...
1
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1answer
19 views

Some basic conceptual question in multivariable partial derivative

Show that $f(x,y) = |xy|$ is differentiable at $(0,0)$. $\lim_\limits{h \to 0} \frac{f(0+h,0)-f(0,0)}{h} =\lim_\limits{h \to 0} \frac{0}{h} = 0$ Similar to partial derivative with respect to $y$ ...
0
votes
2answers
27 views

How to find the limit of a two variable function

Plugging in (0, 0) leads to an indeterminate form (zero divided by zero). Would the only way to approach this problem be through multiplying by the conjugate $\dfrac{(x^2 - y^2)^{3/2}}{(x^2 - ...
6
votes
0answers
88 views

Second Variation of Area Functional

This is a follow up question from this one. I have proved that given a parametrized surface ${\bf x}$, the mean curvature is zero if and only if it is a critical point of the area functional. Then ...
0
votes
1answer
18 views

Finding where tangent plane of ellipsoid intersects x-axis

This is the work I have so far: $f_x = 2x_0,\ \ \ x_0 = x_0 \ \ \ \ \ \ f_x = 2x_0$ $f_y = 6y_0, \ \ \ \ \ y_0 = 0 \ \ \ \ \ \ f_y = 0$ $f_z = 20z_0, \ \ \ \ \ z_0 = 0 \ \ \ \ \ \ f_z = 0 \\$ ...
1
vote
2answers
52 views

I have a ball that intersects a cylinder and I need the volume. How do I do it?

I have an exam coming up and I am stressing out about it really hard. I don't even know how to actually do this. Is it a triple integral? I have a ball $\{(x,y,z)|x^2 + y^2 + z^2 \leq 9\}$ and a ...
1
vote
1answer
48 views

Example 3, Sobolev space Evans

In the following example p.260 Evans. I think I understand everything except for one calculus fact in the second last equation: $$ \int_{\partial ...
0
votes
1answer
21 views

Find a critical point satisfied the Lagrange condition is not local extremum

We know that Lagrange Multiplier gives necessary conditions for an extremum.It locates all possible condidates.But not all such points need be extrma. I want to find an example of the point is ...
1
vote
1answer
28 views

Intuitive explanation of the potential function of a vector field

Suppose I have some vector field $$\vec{f}(x,y)=\begin{pmatrix}A(x,y)\\B(x,y)\end{pmatrix}$$ then the potential function (if the field is conservative) can be found by integrating $A$ with respect ...
1
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0answers
27 views

Finding global maximum

I have a program which must quickly find $x$ and $y$ where $x,y\in\mathbb{N_0}$ which correspond with maximum value of a function: $$f(x,y)=\frac{\sum_{i=0}^{|b|-1}{|b_i ...
1
vote
1answer
22 views

Surface Integral over a Vector Field question

pretty basic question but I can't seem to work it out: Question: Let S be the triangle with vertices $\mathbf{a}=(1,2,3),\mathbf{b}=(1,1,1),\mathbf{c}=(3,1,2)$ with unit normal chosen such that the ...
0
votes
1answer
26 views

What's the best way to think about the Hessian?

I've always thought about the Hessian like this: Let $f:\mathbb R^n \to \mathbb R$ be smooth. Let $g:\mathbb R^n \to \mathbb R^n$ such that $g(x) = \nabla f(x)$. (I am using the convention that ...
0
votes
1answer
24 views

How to go about finding a transformation $T$ in order to solve an integral.

I have the integral $$\int\int_R\left(2x+y\right)dA$$ Where $R$ is the region bounded by $$x+y=-1, x+y = 3, 2x=y,2x-4=y$$ So my first though was drawing the region, which gave me this odd region, so ...
2
votes
1answer
31 views

Order of differentiaton for multivariable functions with arbitrary dependence of variables

While studying Neural Networks, I was bogged with a nasty problem, for which I did not find a satisfying answer using my mathematical knowledge. Let's assume we have a complex multivariable function, ...
0
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0answers
22 views

correction factor

I am trying to figure out a workable formula for temperature compensation Vs load on an aluminum full bridge strain gauge. It seems to be non linear and we are trying to compensate for drift at ...
4
votes
1answer
55 views

Calculation of $A'(0)$ (first variation of the area functional).

I'm trying to do the calculation that shows that a surface in $\Bbb R^3$ is area minimizing if and only if the mean curvature is zero. I'm getting a sign wrong and I'm going crazy, I need help. ...
3
votes
3answers
44 views

What is the unit normal vector of the curve $y + x^2 = 1$

What is the unit normal vector of the curve $y + x^2 = 1$, $-1 \leq x \leq 1$? I need this to calculate the flux integral of a vector field over that curve.
2
votes
2answers
23 views

Deducing a Taylor expansion in an arbitrary point from a MacLauren polynomial

I have a function $f(x,y)=-2x^3 + 4y^3 +4xy+4x$ and I need to find a Taylor expansion, around the point $(-4,1)$ of this function. Since the function is actually a polynomial, I know this ...
5
votes
1answer
37 views

G-P Exercise, immersion except at origin, what does its image look like?

(This is not a duplicate of another question on math.stackexchange, as that other question just basically asks for the answer to the question below, of which I have provided an answer to. My question ...