Tagged Questions

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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0answers
11 views

Double Integral Mistake with Parametric Equation

I'm trying to figure out the mass of an object bounded by $y=0$ and $y=\sqrt{1-x^2}$ the density at a given point is proportional to its distance from the origin; $\rho(x,y) = kxy$. So I set it up ...
0
votes
2answers
11 views

Gradient of a vector field

We did in lectures gradient of a scalar field and I am wondering how is the grad of a vector field. I tried the following Let V=f(x,y,z)i+g(x,y,z)j+h(x,y,z)k Then by definition grad(V)=iVx + jVy +kVz ...
1
vote
0answers
8 views

Affine functions as equality constraints in convex optimization problems

I am studying on an introduction to convex optimization problems. When defining a convex optimization problem, we have a convex object function, $f(x)$, a set of convex functions $g_i(x)$ where the ...
0
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0answers
20 views

nice name for the image of multivariable function

Consider a differentiable function $f:D\subset\mathbb R^m\mapsto \mathbb R^n$ with $m\le n$. I know if $m=1$ then $f(D)$ is called by "path", if $m=2$ then $f(D)$ is called by "surface" and if $m=3$ ...
0
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0answers
15 views

Why did my teacher integrate φ from (π/3) to (π/6) while solving a triple integral for volume in spherical coordinates?

In Calculus III class today we learned how to evaluate triple integrals in spherical coordinates. One of the example questions we worked on was to use a triple integral to find the volume of a shape. ...
0
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2answers
30 views

How integrate $ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $

I'm trying to resolve this integral $$ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $$ I tried with polar coordinates: $$ x = r\cos{\theta} \\ ...
1
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2answers
16 views

solve polar coordinate integral

Evaluate $$\int_0^R\int_0^\sqrt{R^2-x^2} e^{-(x^2+y^2)} \,dy\,dx$$ using polar coordinates. My answer is $-\frac{1}{2}R(e^{-R^2+x^2}-1)$ but I want to confirm if that's correct And also, when I ...
1
vote
1answer
46 views

Show that f solves the so called wave equation

Task $\text{Let } \; c \in \mathbb{R} \; \text{ be a given parameter, with } \; c > 0$ $\text{ Show that } \; f: (\mathbb{R}^3 \setminus \{ \vec{0} \}) \times \mathbb{R} \to \mathbb{R} \; ...
0
votes
1answer
23 views

Prove that $f(x,y) = 10x^2+ 10y^2 + 12xy + 2x + 6y + 1 \geq 0$

I need to prove that $f(x,y) = 10x^2+ 10y^2 + 12xy + 2x + 6y + 1$ has a global minimum at the point $(1/8, -3/8)$. Since $f(1/8, -3/8) = 0$, that would be proving that $f(x,y) \geq 0$ $\forall ...
0
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0answers
23 views

Surface integral defined by a closed curve

So I know how to integrate over a surface defined by a parametric equation $$ \textbf{r}(u, v) = x(u, v) \textbf{i} + y(u, v)\textbf{j} $$ But what if the surface is defined as the area inside a ...
1
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0answers
22 views

Gauss-Green Theorem from generalized Stoke's Theorem.

I am trying to deduce the next identity (Green-Gauss theorem) $$\int_\Omega \dfrac{\partial u}{\partial x_i} dx = \int_{\partial \Omega} uv_i dS$$ from the generalized Stoke's theorem for manifolds. ...
0
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1answer
17 views

Finding the optimal combination for the Cobb-Douglas function given a budget

I am trying to figure out to find the optimal combination of the Cobb-Douglas function given some budget. An example question is: Output can be produced with labour and capital according to $Q = ...
-1
votes
3answers
35 views

Find $\lim \limits_{x,y)\to (0,0)}\left(\frac{\sin(x+y)}{x+y}\right)$ [on hold]

Let $f(x,y) = \dfrac{\sin(x+y)}{x+y}$. Does this function have a limit when $(x,y)\to {\bf0}$?
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0answers
5 views

Visualizing construction of a certain function

Consider a map $f$ defined on $\mathbb{R}^3$ with the following properties:\ 1) $f$ fixes the poles $(0,0,\pm1)$.\ 2) $f$ is symmetric in the plane $\{x=0\}$ and the plane $\{y=0\}$.\ 3) $F$ is ...
2
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0answers
19 views

Find the initial movement of a particle

A particle with mass $m$ is moving along a curve and the force exerted on it always points towards the origin, and it´s magnitude is proportional to the distance between the particle and the origin, ...
0
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2answers
25 views

How to evaluate this path integral?

So I know that the integral is $$\int_1^e f(x(t), y(t), z(t))(||c'(t)||) \:dt$$ I set this to$$\int_1^e\frac{1}{t^3}\sqrt{\frac{1}{(ln10*t)^2}+1}\; dt$$ I found this too hard to integrate by hand, ...
0
votes
1answer
21 views

Find the mass of the disk. - Double Integration Problem - Calculus 3

A disk of radius 5 cm has density 10 g/cm2 at its center, density 0 at its edge, and its density is a linear function of the distance from the center. Find the mass of the disk. my answer: 157.08g ...
2
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1answer
35 views

Double Integral: finding the area using a parametric equation

$$ r = \cos (3\theta) $$ I need to find the area of one loop of the rose made by this function. I know that the bound for $\theta$ is $-\pi/2$ to $\pi/2$ for one of the loops. Although I don't know ...
3
votes
1answer
50 views

Multiple integral differential notation

When writing a multiple integral, I have noticed there is sometimes used a shorthand for writing the differential in the integral. For example in $\mathbb{R}^3$ instead of writing $\mathrm{d}x\ ...
0
votes
1answer
23 views

How to find this maximum?

How to find the maximum of $$f = |x_1x_3 + x_1x_4 + x_2x_3 -x_2x_4|$$ on the four-dimensional cube $\{x \in \mathbb{R}^4:|x_k| \le 1, 1 \le k \le 4\}$? Calculations with CASes suggest it equals 2.
3
votes
2answers
83 views

Computing the Laplacian of $\frac{\mu\ \cdot\ \mathbf{r}}{\|\mathbf{r}\|^3}$

How does one compute the Laplacian of: $\dfrac{(\mathbf \mu \cdot \mathbf r)}{r^3} \;\; \text{where} \;\; r = \Vert \mathbf r \Vert$ I am aware that the Laplacian is defined: $\sum ∂_i^2f$ but am a ...
2
votes
1answer
32 views

maximization using Lagrange

I am maximizing $f(x,y)=-x$ given the constraint $g(x,y)=x^2-y^2=0$ To satisfy the non degenerate constraint qualification I have: $Dg(x,y)= [2x\quad-2y]$ and the set of $(x,y)$ that satisfy it ...
1
vote
3answers
73 views

Solve $ \int_0^{\sqrt{\pi / 2}}\left(\int_x^{\sqrt{\pi / 2} }\sin(y^2) dy \right)dx$

I'm trying to solve this: $$ \int_0^{\large\sqrt{\frac{\pi}{2}}}\left(\int_x^{\large\sqrt{\frac{\pi}{2}}}\, \sin y^2\, dy \right)dx $$ But I'm having trouble with finding an primitive to $\sin(y^2)$. ...
0
votes
0answers
25 views

Integral of a radially increasing function

Fix a radius $R > 0$, a positive integer $n$ and suppose $B(s, R) \subset \mathbb{R}^{d}$. Let $f(s) := \int_{B(s, R)}|x|^{n}\, dx$. Is $f$ monotonically increasing in $|s|$? That is, suppose we ...
0
votes
1answer
13 views

Existence of a unique maximizer of a strict quasi-concave function defined over a convex set

Set $S \subset \mathbb R^2$ is compact and convex. A typical element of $S$ is $s=(s_1,s_2) \in S$. Also, $d \in \mathbb R^2$ is a fixed element such that there exists $s \in S$ such that $s \gt d$. ...
1
vote
1answer
26 views

Second derivative test failed

I have $$H(\psi_1(t),\psi_2(t),x_1(t),x_2(t),u(t))= -1 + \psi_1(3x_1+x_2) + \psi_2(4x_1+3x_2 +u)$$ Note: $\psi_1 = -2Ae^t + 2Be^{5t}$, $\psi_2 = Ae^t + Be^{5t}$ Hence we have $$\frac{\partial ...
0
votes
1answer
23 views

Is there free software that can be used to generate a chain rule tree graph?

I'm in multivariate calculus and we just finished up the chain rule. One of the methods for solving this is to produce a tree graph and traverse it. An example would be this tree graph taken from my ...
0
votes
2answers
42 views

Differentiating $\langle Ax,x\rangle$

If $f\colon\Bbb R^n\rightarrow\Bbb R^m$ and $g\colon\Bbb R^n\rightarrow\Bbb R^m$ are differentiable at a point $x_0\in\Bbb R^n$, and $F(x)=\langle f(x),g(x)\rangle$, then ...
1
vote
1answer
32 views

find critical point $f(x,y) = 8 + 2y^3+2x^3 - 3xy$

$f(x,y) = 8 + 2y^3+2x^3 - 3xy$ $f_x = 6x^2 - 3y$ $f_{xx} = 12x$, $x= 0$, $y=0$ $f_y = 6y^2 -3x$ $f_{yy}=12y$ $f_{xy}=-3$ at $(0,0)$, $D = -9 < 0$ Is it one saddle point $(0,0)$?
3
votes
1answer
25 views

Triple integral over an ellipsoid

Let $E$ be the solid ellipsoid $E = ${$(x,y,z)$ | $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \le 1$} where $a > 0,\: b > 0,\: c > 0$ Evaluate $\int\int \int xyz\: dxdydz$ over: a. ...
0
votes
4answers
35 views

Finding maximum area of rectangle with constraint

Find the maximum area of a rectangle in the xy-plane with its sides parallel to the axes, one vertex at the origin, and the diagonally opposite vertex on the curve $$ x^2 + y = 1 $$ I am supposed ...
0
votes
1answer
15 views

Getting the points at which the tangent plane is horizontal?

Given the equation: $$ z = xy(3-x-y). $$ Find all the points where the tangent plane is horizontal. I found $Fx = Fy = 0$ and got to the following system: $$ \begin{cases} 3x - x^2 - xy = 0\\ ...
0
votes
1answer
18 views

Transformation to elliptical coordinates

I'm currently struggling to make any progress with this question. I'm a little bit thrown by the inclusion of cosh and sinh. I am aware of all of the definitions, just need guidance with approach.
0
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2answers
16 views

Convert a triple integral to cylindrical coordinates?

Find the volume determined by $$z \le 6-x^2-y^2$$ and $$z \ge \sqrt{x^2+y^2}$$ I used cylindrical coordinates to change the bound for $z$ to $r \le z \le 6-r^2$. However, I am not sure how to find ...
0
votes
2answers
25 views

Finding the bounds on a triple integral

Problem: Find the volume enclosed by the cone $$x^2 + y^2 = z^2$$ and the plane $$2z - y -2 = 0$$ So I know that I need to do a triple integral over this region, and the integrand will be 1. My ...
0
votes
1answer
29 views

Derivative of $F : \mathbb R^{n+m} \rightarrow \mathbb R^{n+m}$

Question : Let f be continously differentiable function on an open set E $\subset \mathbb R^{n+m}$ into $\mathbb R^n$. Define a funtion F on E into $\mathbb R^{n+m}$ such that F(x,y) = ...
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votes
2answers
18 views

which chain rule

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ of $z=f(\frac{x}{y})$ Thoughts If I write $g(x,y)=\frac{x}{y}$, then $z=f(g(x,y))$. Which Chain Rule do I use now? help ...
0
votes
0answers
7 views

Alternating Multilinear Functions

I'm watching this video, it's a course about differential forms. In the video, (time is ~08:00-08:13), it says: for $v_1 = \begin{pmatrix} 1\\2\\4 \end{pmatrix}$ and $v_2 = \begin{pmatrix} ...
1
vote
2answers
47 views

Double Integration with change of variables

I am having trouble with the following double integral: $$\iint\limits_D(x^2+y^2) \;dA$$ where $D$ is given by the region enclosed by the curves $xy=1$ $xy=2$ $x^2-y^2 =1$ $x^2-y^2 =2$ I have ...
0
votes
0answers
28 views

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces?

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces (explicitly)? A derivation is here but its conclusions seems not right? The expected ...
1
vote
1answer
15 views

Evaluate the Integral with 2 Paths

I'm guessing I will have to split this integral into 2. One with a circle and another one with a line. Then add them up. How do I evaluate such a thing? What does this integral mean? Is the the ...
1
vote
1answer
55 views

Beginner exercise on the Implicit function theorem

(I'm german and so I'm struggling to express my problem in the right technical terms, I'm temporarlily trying to translate this with google translator and wikipedia lookups - I'd need a hand there) ...
1
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0answers
17 views

Applying directional derivatives

A spaceship is at location $(1,1,1)$ and the temperature of the ship's hull when at location $(x,y,z$ will be $$ T(x,y,z) = 200 +e^{-x^2-2y^2-3z^2} $$ where x,y,z are in meters. a) In what ...
4
votes
1answer
41 views

Find K values that make a differential equation solution stable

Given some differential equations, ie. "a", or "b": a. $$Y'''+Y''+2Y'+KY=0$$ b. $$Y'''+KY''+3KY'+2Y=0$$ How do I get the $K$ values that make the solution stable? I know that for "a", it should be ...
3
votes
3answers
41 views

Multivariable limit … no L'Hopital rule?

I am looking a bit at limits for multivariable functions by myself, and I can't figure it out; my book only mentions them shortly, but now I am looking at an "assignments for those interested" and it ...
1
vote
1answer
20 views

Find the arclength parametrization of the curve

Find the arclength parametrization of the curve $$r(t) = (t^2, t^3 ), t>=0$$ I find the integral of |r'(t)| = ${1\over9} \left( {(4+9t^2)^3\over 3} -{4^3\over 3}\right)$ The integral looks ...
3
votes
1answer
44 views

Proving the relation: $∇(\mathbf{u}·\mathbf{v})=(\mathbf{v}·∇)\mathbf{u}+(\mathbf{u}·∇)\mathbf{v}+\mathbf{v}×(∇×\mathbf{u})+\mathbf{u}×(∇×\mathbf{v})$

I have to prove the following relation. I am looking for a solution beyond the obvious brute force method of considering ...
0
votes
0answers
26 views

Help solving ODE using Laplace Transform

The problem: $y '' +4y ' + 40x = 0$, seems odd to me because of the $x$ variable. I don't know if this is a typo or not. Can you please assist?
1
vote
2answers
29 views

Continuous differentiability of atan2

Consider the function atan2 defined on the plane, minus the origin and the negative $x$-axis, as the unique $\theta$ such that $-\pi<\theta<\pi$ and $$ x = r \cos \theta, \qquad y=r \sin ...
0
votes
2answers
22 views

Show that $\boldsymbol{\mathrm{F}}$ is independent of path.

Consider a vector field $\boldsymbol{\mathrm{F}}(x,y) = \langle 2xy, x^2 \rangle$ and three curves that start at $(1, 2)$ and end at $(3,2)$. Explain why ...