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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2answers
25 views

Find the point on the ellipse where the cylinder intersects the plane furthest from the origin?

I'm confused about how I should set this problem up. It is a lagrange problem. The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from ...
22
votes
6answers
4k views

What does a triple integral represent?

From my understanding if the integrand is 1, then it gives you the volume of the region defined by the bounds. But what does the value of a triple integral represent if the integrand is a function for ...
1
vote
1answer
24 views

Area of Region using Transformation

Let R be the region bounded by the curves x = 0, y = sin(x)+1, y = sin(x), and y = 2 − x. Find the area of R. I need to use a transformation to find this, but I could not solve it using a ...
0
votes
1answer
33 views

Proving ${\displaystyle{\int\!\!\int_{D}\!\!u\Delta udA<0}}$ where $D$ is the closed, unit disc.

Hello again guys and gals! I'm stuck on a problem that I thought was going to be simple for me to prove, but I was wrong (yet again). I will try not to be so long-winded this time (see my previous ...
3
votes
1answer
82 views

What is the most general way to think about Integrals?

Given a single-variable scalar function, $f : \mathbb{R} \to \mathbb{R}$ The "area under the curve" (of the graph of the function $f$ in $\mathbb{R^2}$) is given by $$\int_{a}^{b} f(x) \ dx = ...
0
votes
1answer
42 views

Double integral of an absolute valued function (in polar coordinates)

I'm having a problem separating the integral Calculate the integral over given region by changing to polar coordinates $$f(x,y)=|16xy|,\quad x^2+y^2\le 25.$$ I'm not really sure where to ...
1
vote
1answer
32 views

Find the global extrema of $f(x,y)=\sin(xy)$ on $D=[(x,y)| x = [0,\pi], y=[0,1]]$

Find the absolute maximum and absolute minimum of the function: $$f(x,y) = \sin(xy) \text{ on } D=[(x,y)| x = [0,\pi], y=[0,1]]$$ I took the partial derivatives and got: $$\frac{df}{dx} = \cos(xy)y ...
0
votes
1answer
12 views

Computing line integral

Can someone please help me solve the following line integral? I have difficulty with heuristics of these kinds of problems, any help would be appreciated. Let $F(x,y)=(x^2,-y),$ $C$ is the graph of ...
0
votes
0answers
10 views

Vector Calculus Divergence Theorem Textbook Answer Confusion

here's a particular question I'm working on that the textbook doesn't have the same answer as me. Use The Divergence Theorem for: $F = |r|r$, where $r = <x,y,z>$, and $S$ consists of the ...
0
votes
1answer
31 views

Bounding inequalities in three dimensions

I want to write $z^2 \ge x^2 + y^2$, $x^2 +y^2 +z^2 \le 1$ and $z \ge 0$ in the form $$a \le z \le b, \quad c(z) \le y \le d(z), \quad f(y,z) \le x \le g(y,z)$$ or $$a \le z \le b, \quad c(z) \le ...
0
votes
0answers
11 views

What is this operator? (Three times curve integral)

What is this operator: https://help.libreoffice.org/File:Fo21611.png I have been seeing it in text-edit documents, but never found any explanation to it. I guess that it is a closed curve integral ...
1
vote
0answers
13 views

Evaluation of double integral

Find $\int\int_T xy \, \mathrm{d}A$ where T is the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$. It's really a question about the bounds and not so about evaluating the actual integral. I ...
-1
votes
2answers
23 views

Limit of a function in two variables

The problem: Find the limit as $(x,y)$ tends to $(2,2)$ for the following function: $$f(x,y)= ⁡\frac{y^2-2xy+x^2}{x-y}.$$ Not sure what to do for this question, how would I go about solving?
0
votes
0answers
21 views

How to find $\partial ^2 u/ \partial x^2$?

If we have $u=u(\xi , \eta )$ and $\xi=\xi (x,y)$, $\eta = \eta (x,y)$ and $$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x} +\frac{\partial u}{\partial ...
0
votes
0answers
31 views

Elementary properties of diff. form from PMA Rudin

How he got $d\omega$ in the RHS of $(39)$? Or maybe it's a typo?
0
votes
0answers
17 views

Why can't I seem to reconcile constants in my Surface Integral Formula?

I was trying to re-derive the surface integral formula from scratch and I came to a problem involving constants. I begin by considering a point A: $(x,y,z(x,y))$ and an infinitesmal change in the x ...
-1
votes
1answer
43 views

Solve $\iiint \limits_{0<x<1,~0<y<1,~0<z<1,~x+y+z<2} xyz \,dx\,dy\,dz$ [closed]

Please help me solve this definite integration problem: $${\iiint \limits_{0<x<1,~0<y<1,~0<z<1,~x+y+z<2} xyz \,dx\,dy\,dz}$$
1
vote
1answer
21 views

Justification of this step involving Gaussian Integral

I have came across the following step, I suspect it is true but I am not sure how it is justified. $$ I = \int_{-\infty}^{\infty} \frac{dk}{2\pi} \exp{\left(i k (x_0 - x_t) - \frac{k^2}{2} ...
0
votes
2answers
15 views

What is the equation for the walls of a 3D cylinder?

If for example I have the circle $x^2 + y^2 = 4$ in the $x$-$y$ plane, and I want to extend it upwards into the $z$ dimension, how would I write the equation for the circular walls in terms of $z$?
2
votes
2answers
49 views

Sum of a step function

Please help me to find the following summation $$\sum_{m=0}^{N} \sum_{n=0}^N \sum_{i=0}^N \sum_{j=0}^N \mathbb{1}(m,n,i,j)$$ where $1(m,n,i,j)= \left\{ \begin{matrix} 1 \qquad m+n=i+j \\0 \qquad \; \; ...
0
votes
1answer
17 views

Compute the the tangent line for the curve obtained by intersecting two surfaces

Let C be the curve obtained by intersecting $x^3 + 2xy + yz = 7$ and $3x^2 - yz = 1$. Find the parametric equations of the tangent line to C at $P = (1,2,1)$. So I first set the equations equal to ...
4
votes
1answer
53 views

Differentiation under integral sign?

I am trying to understand the following argument given in a text book: Suppose $f \in L^1(\mathbb R^n)$, consider the function $\hat{f}(\zeta)= \int_{\mathbb R^n} \exp(-2\pi i X.\zeta)f(X)dX$. ...
0
votes
1answer
23 views

Compute volume of region

I need to compute the volume of a region bounded by the planes $z=0$, $y=-1$, $y=1$ and the parabolic cylinder $z=1-x^2$. I really don't know how to tackle this problem. I thought maybe something ...
0
votes
1answer
52 views

The shape of the hill is given by given by the function. If you walk due to south-west direction, will you start to descending or ascending?

The shape of the hill is given by given by the function $$f(x,y) = 5000 - 0.001x^2 - 0.04y^2$$ Where $x$ and $y$ are measured in meters, and the positive $x$-axis points east; positive $y$-axis, ...
0
votes
0answers
10 views

Calculating the Jacobian for a linear transformation (Kadane's 'Principles of Uncertainty') V2.

Let $A$ be a symmetric positive definite matrix, and $T$ a lower-triangular, with $A=TT´$. In this book, page 316 at the bottom, when trying to find the Jacobian of the transformation A, the author ...
1
vote
2answers
35 views

Volume of the region in $\mathbb{R}^3$ defined by $z^2 \ge x^2 + y^2$, $x^2 +y^2 +z^2 \le 1$ and $z \ge 0$

I want to find volume the region in $\mathbb{R}^3$ defined by $z^2 \ge x^2 + y^2$, $x^2 +y^2 +z^2 \le 1$ and $z \ge 0.$ I set it up as $\displaystyle ...
1
vote
1answer
24 views

Evaluation of $\oint_C y(y-x^2)\, dx +x (1+y)^2 \, dy$ where $C$ is the closed curve $x^2+y^2 = 4$.

I want to evaluate $$\oint_C y(y-x^2)\, dx +x (1+y)^2 \, dy ,$$ where $C$ is the closed curve $x^2+y^2 = 4$. I want to know why the solution below using Green's Theorem is wrong: ...
0
votes
0answers
19 views

Find the equivalent iterated integral.

Given : $$\int_0^1 \int_0^{1-x^2}\int_0^{1-x} f(x,y,z) \,dy dz dx $$ I need help with this integral, since there is nothing in yz plane so I solved both equations for y and z. My attempt:(Is it ...
0
votes
1answer
32 views

Calculating the Jacobian for a linear transformation (in Kadane's 'Principles of Uncertainty')

Let $Y=BXB^T$, where $B$ is non-singular, and $X$ is a symmetric non-sigular $n\times n$ matrix. According to the book Principles of Uncertainty, on page 316, if $B$ is a matrix of type ($i$), just ...
1
vote
0answers
14 views

Average Value of a Line Integral

I'm having quite a hard time calculating the average value of a line integral. Given the surface $f(x,y) = \sqrt{16 + 36y^{2/3}}$ and the curve $y = x^{3/2}$, I need to calculate the average value of ...
1
vote
0answers
21 views

Two multi-variable limit problems

$$\lim_{(x,y)\to(0,0)}\frac{2x^2y}{x^4+3y^2}$$ I'm getting that the limit DNE because using $(0,y)\to(0,0)$ it is $0$ but for $(x,x^2)\to(0,0)$ it is $1/2$. Since $0$ does not equal $1/2$ the limit ...
2
votes
1answer
41 views

Show that $\int_{0}^{\infty} \int_{0}^{\infty} e^{-s(x+y)}f(x)g(y) \ dx \ dy = \int_{0}^{\infty} \int_{0}^{t} e^{-st}f(t-u)g(u) \ du \ dt$.

While learning how to compute the product of two Laplace transform and the inverse transform, I faced with this equality : $$\int_{0}^{\infty} \int_{0}^{\infty} e^{-s(x+y)}f(x)g(y) \ dx \ dy = ...
0
votes
0answers
17 views

Is it d^3V or dV when integrating for volume (triple integral)?

In a volume integral dxdydz is shortened to dV. I have also seen d^3V. So dcubed V. What is the correct way to write it, or are both correct?
-1
votes
1answer
18 views

Torque when system is constrained to rotate about $\vec{r}$ [closed]

Let $\vec{F}, \vec{r}$ and $W$ are elements of $R^{3}$. Given $\vec{F}= -\nabla{W}$. Let system be constrained to rotate about $\vec{r}$. How can we find Torque?
0
votes
0answers
44 views

derivative of inverse of matrix

This is problem of Marsden's elementary classical analysis Question: Consider the map $L^{-1} $ : $GL(\Bbb R^n, \Bbb R^n)$ $-> $ $GL(\Bbb R^n, \Bbb R^n)$ , $A$ $\mapsto$$A^{-1}$ taking a ...
0
votes
1answer
33 views

Limits with more than 1 variable

For $\lim \limits_{(x,y,z)\to(0,0,0)}\frac{x+y+z}{x^2+y^2+z^2}$ are the following correct? $\lim \limits_{(x,y,z)\to(0,0,0)}\frac{x+y+z}{x^2+y^2+z^2}$ for $x=y$ and $x=z$ $\lim ...
1
vote
2answers
58 views

inverse of the function $f(r,\theta) = (r\cos \theta, r \sin \theta)$

inverse of the function $f(r,\theta) = (r\cos \theta, r \sin \theta)$ set $x = r \cos \theta$, $y = r \sin \theta$ then we have $ x^2 + y^2 = r^2$ so $r = \sqrt{x^2+y^2}$. Now $y/x = \tan \theta$ so ...
1
vote
1answer
15 views

Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
0
votes
1answer
20 views

Simplifying second order chain rule

In these notes: http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx We have $$\text{If } y = f(x(t)), x = g(t)$$ $$\text{Then } \dot y = \dot{f}(x(t)) =\dfrac{ \partial f}{\partial ...
0
votes
1answer
25 views

Finding the critical points of a trigonometric function

Question: Find the critical points of $f(x,y) = \sin(x)\sin(y)\sin(x+y)$ with the open domain $D=$ {$0\lt x \lt \pi, 0 \lt y \lt \pi$} Answer: I have found that there is a critical point ...
1
vote
1answer
42 views

Finding the maximum and minimum values of the function $u(x,y)=e^x \cos y$.

Let $D$ denote the unit disk centered at the origin. I'm trying to find the maximum and minimum values of the function $$u:D \to \mathbb{R},\,\ (x,y) \mapsto e^x \cos y.$$ I'll try using Lagrange ...
0
votes
1answer
14 views

Finding the radii of an ellipse from the intersection of a plane and a sphere

I'm trying to solve the following problem, regarding Stokes Theorem: $F = z i + xj + yk $; C the curve of intersection of the plane $x + y + z = 0$ and the sphere $x^2 + y^2 + z^2 = 1$ [Hint: ...
0
votes
1answer
35 views

Find arc length of $(x-1)^{2/3}+(y-2)^{2/3}=1$

I'm trying to find the arc length of the curve defined by $$(x-1)^{2/3}+(y-2)^{2/3}=1$$.My first approach was try to set 'y' in terms of 'x' and then apply the formula ...
0
votes
0answers
18 views

Line integrals and finding the mass using the latter

I need help validating my work, there are no answer keys and I am very rusty with line integrals. I feel like I'm doing something very wrong. Here's the question(It's downloaded , past final exams): ...
0
votes
1answer
26 views

Evaluate the following line integral.

I have to find the line integral of the following. $$\int_Cxe^ydx+x^2ydy, C: 0 \leq x \leq 2, y = 3$$ I am trying to understand the concept of line integrals, but in this case, I am confused as to ...
1
vote
1answer
38 views

lebesgue and riemann integrals are the same for continuous functions on $[a,b]$

I have a proof in front of me which goes as follows, firstly assuming that the function $f \geq 0$ on $[a,b]$. We get a partition $a = x_0 < x_1 <....<x_n = b$ with $x_i - x_{i-1} = ...
0
votes
1answer
21 views

Double integral Clarification

Evaluate the double integral. $$\iint (x^2 − xy) dA$$ $R$ is the region enclosed by $y=x$ and $y=3x-x^2$ A little lost. My approach $x=0, x=3 y=x, y=3x-x^2$, $\iint(x^2-xy) dydx$. I end up with ...
1
vote
1answer
33 views

Boundary of unit sphere [duplicate]

Let $S$ be a unit sphere in $\mathbb{R}^3$ - I am told that $\partial S = \emptyset $, but why does the unit sphere have no boundary?
1
vote
0answers
17 views

Solving for critical points and classification using Maple

I am trying to help a friend on an assignment using Maple. The question gives a function $$f(x,y,z)= x+\frac{y^2}{4(x-5)}+\frac{z^2}{y-3}+\frac{2}{z+5}$$ and asks to find and classify the critical ...
0
votes
0answers
38 views

Is the electric field of a volume charge distribution well defined?

Consider a volume charge distribution $\rho({\bf r'})$. The electric field at ${\bf r}$ is $${\bf E}({\bf r})=\iiint\frac{1}{4\pi\epsilon_0}\frac{\rho({\bf r'})}{R^2}\hat{\bf R}d^3{\bf r'}$$ where ...