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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65 views

What is the potential function of the field $\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$

The vector field is obviously conservative on every closed domain that doesn't encompass the point $(0,0)$, so there must be a potential function. I've got $\arctan(\frac{x}{y})$ for $x$ unequal to ...
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0answers
24 views

What order should I evaluate divergence and coordinate transformation if I want to use a different coordinate system?

I have a vector field in Cartesian coordinates. I need to find its divergence, but I need the divergence to be in spherical coordinates. However, the divergence of this field is far easier to ...
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1answer
34 views

Trigonometric Partial Derivative

I need to find $$\frac{\partial Z}{\partial U} \text{ and } \frac{\partial Z}{\partial V}$$ for a $z=f(x,y) = \cos(xy) + y\cos(x)$. After a bit of an internet search, I think I have found the ...
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1answer
34 views

Partial Derivative Stickler.

I am having trouble with a question with partial derivatives. Here is the question: Let $\rho = \sqrt{x^2 + y^2 + z^2}$ Show that $$\frac{\partial ^2\rho}{\partial x^2} + \frac{\partial ...
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0answers
42 views

Theorem 4.6 in Spivak's Calculus on Manifolds

Could you elaborate on the proof please? This is how I would prove the theorem: Since $\Lambda^n(V)$ is $1$-dimensional, $\omega=\alpha(\phi_1\wedge\phi_2\wedge...\wedge\phi_n)$ for some $\alpha$ ...
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2answers
42 views

Having a bit of trouble with min/max distance from sphere to point

The sphere is $x^2 + y^2 + z^2 = 81$ and the point is $(5,6,9)$ I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon. I found ...
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1answer
35 views

Vectors with given angle and magnitude

Give an example of vectors $\mathbf{v}$ and $\mathbf{w}$ such that the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\frac{2\pi}{3}$ and $\|\mathbf{v} \text{ x } \mathbf{w}\|=\sqrt{3}$. Should I ...
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0answers
50 views

Paraboloid Curvature calculation methods

If we have a paraboloid generated as a surface of revolution of the 2d function $f(x)=ax^2+b$, the equation of the 3d graph is $f(x,y)=ax^2 + ay^2+b$. The gaussian curvature of a 3d graph $f(x,y)$ is ...
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2answers
125 views

Solving a particular system of differential equations

The problem I'm trying to solve is this: $X'(t) \in \mathbb{R}^3 \,, \, \omega = (\omega_1,\omega_2,\omega_3) $ Find the general solution for $$X'(t) = \omega \times X(t)$$ After doing the cross ...
1
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1answer
39 views

Cobb Douglas Difficulty

Show that the Cobb-Douglas production function, for Labour costs L and Capital costs K, $P(L, K) = AL^{\alpha}K^{1-\alpha}$ satisfies the equation: $$L\frac{\partial P}{\partial L} + ...
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1answer
38 views

Need Help Understanding How To Integrate With An Implicit Variable

My calculus is really rusty (damn Mathematica/Matlab) and I was wondering if anyone could help me with an equation I am having trouble integrating. I have attached a snapshot of the paper I am trying ...
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0answers
36 views

A question regarding calculus in several variables.

Let $f:\Bbb{R^n}\to \Bbb{R}$ be a continuous function. I refer to this proof of the Mean Value Theorem for several variables. Here $g(t)=f(a+t(b-a))$. Hence ...
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1answer
47 views

Taking Fourier transform of integral-differential equation

If $u$ is a solution of the equation $$\frac{\partial}{\partial t} u(x,t) + \int_{-\infty}^{\infty} \text{sinc}(x-y) \cdot \frac{\partial^{2}}{\partial y^{2}} u(y,t) \ dy = 0,$$ with initial condition ...
0
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1answer
102 views

Transform square region to triangular region

How do you express x and y in terms of u and v so that the region $\{(u,v): 0\le u, v\le 1\}$ is mapped to the triangular region in the $xy$-plane with vertices $(0,0)$, $(1,0)$, and $(0,1)$? Now, ...
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0answers
51 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
2
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1answer
116 views

Explanation of Lagrange Equation with Chain Rule

I am just reading through some lecture notes explaining the Lagrange Equation, and I am a bit confused with some chain rule stuff, I get to the part with: $$\frac{\partial F}{\partial y} = ...
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1answer
34 views

How to calculate the partial derivatives of the composition $F(u(s,t),v(s,t))$?

Could someone help me to understand how to do this problem? I believe Partial Derivatives are used. Thanks!
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2answers
55 views

Shortest distance between a point an a non linear surface

I am trying to find the shortest distance between a point $P(p_1,p_2,p_3)$ and the surface $z=xy$. Could somebody help me please?
0
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1answer
30 views

Rewrite the following surface so that I can graph it.

$z = \dfrac{1+x^2}{1+y^2}$ $ $ I want the part of the surface above the square $|x|+|y|\leq 1$ $ $ OR we can write this square as $-y<x<y$ and $-1<x<-1$ $ $ I have spent hours trying ...
2
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2answers
84 views

Double integral where limits are the first quadrant

Evaluate the integral $$\iint\limits_D \frac{1}{(x+y+1)^3} \, dA$$ where $D$ is the first quadrant. In this case, what would the limits of integration be? I'm having trouble moving to polar ...
4
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1answer
170 views

Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit ...
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1answer
35 views

Diagonal matrices and integrals

Suppose that $$A=\int_{\alpha}^{\beta} f(B,x)\ dx,$$ where $B$ is a $3\times3$ matrix. The result I'm looking for is that if $B$ is diagonalized with an orthogonal matrix, then is A diagonalized by ...
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2answers
44 views

Is this proof of multivariable limits legit?

Show that the limit $\lim \limits_{(x,y) \to (0,0)} \frac{x^2 - y^2}{x^2 + y^2}$ does not exists. Step 1) let $x=0$ $\Rightarrow \frac{-y^2}{y^2} =1$ Step 2) let $x = y \Rightarrow \frac{0}{2y^2} =0 ...
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1answer
47 views

Work done by a force field line integrals

Find the work done by the force field $F(x, y) = \langle 2x \sin(y), 2y \rangle$ on a particle that moves along the parabola $y = x^2$ from $(-1, 1)$ to $(2, 4)$. So to use line integrals to solve ...
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1answer
77 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
2
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1answer
50 views

Computing double integral

Find $$\iint\limits_D \sqrt{(x-10)^2+y^2}\hspace{1mm}dA$$ where $\{(x, y)\in D \mid x^2+y^2\leq 10^2\}$. I am not sure how to start, every way I have tried so far, ends up into something ugly. All ...
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1answer
38 views

Finding a function from a vector field

The vector field $F(x, y) = \left(\displaystyle\frac{x}{r^3}, \frac{y}{r^3}\right)$ appears in electrostatics, where $r = \sqrt{x^2 + y^2}$ is the distance to the charge. Find a function $f(x, y)$ ...
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270 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
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2answers
90 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
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2answers
60 views

Verification of the Stokes theorem for the surface that is a part of a cone

Let $S$ consist of the part of the cone $z=(x^2+y^2)^{1/2}$ for $x^2+y^2\leq9$ and suppose $${\bf A}=(-y,x,-xyz).$$ Verify that Stokes theorem is satisfied for this choice of $\bf A$ and $S$. In ...
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0answers
18 views

Trace of hyperbola $2$ sheets

For $yz$ parallel trace, the book mentions that there is no trace unless $|x|>|a|$, in which case the trace is an ellipse. It is hard for me to visualize this. Any insight would be appreciated, ...
0
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2answers
141 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
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1answer
82 views

Bounding the integral of a function by the integral of its derivative

I have no idea where to begin for this question, so any help would be greatly appreciated! Let $\Omega$ be a square with side 1. Show that $$\left(\int_{\Omega} v^2 \, dx \right)^{1/2} \leq \left( ...
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0answers
64 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
3
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1answer
63 views

Sketching a surface

If $${\bf F}=2y{\bf i}-z{\bf j}+x^2{\bf k},$$ and $s$ is the surface of the parabolic cylinder $y^2=8x$ in the first octant, bounded by the planes $y=4$ and $z=6$, evaluate $$\int_S{\bf ...
0
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1answer
188 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
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3answers
112 views

How do I find a point on the surface of a sphere

How do I find a point on a sphere knowing its radius and center point ? I have a sphere: $$x^2+(y-1)^2+(z+3)^2=16$$ Obviously its center point is $(0,1,-3)$ and its radius is $4$. I am asked to find ...
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2answers
35 views

Parametrization Question

When computing a line integral, or any integral that requires parametrization, what are you integrating with respect to? For example, if parametrizing in polar coordinates, with $x=r\cos\theta$ and ...
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1answer
29 views

Average value for multiple integrals

If there is a function $f(x,y)$ and we want to find the average value over a region $R$ defined by $0<x<1$ and $0<y<x$, how is that computed? I know that it would be something like this: ...
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1answer
44 views

Minimization with two functions that are not completely related

Two caveats: 1) This is a problem I formulated myself, and so may not be structured correctly/logically. 2) I don't have an extensive math background, but am currently finishing up Calc 3. I have an ...
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1answer
67 views

Use triple integrals to integrate over a tetrahedron

Integrate $f(x, y, z) = x^2 + y^2 - z$ over the tetrahedron with vertices $(0, 0, 0), (1, 1, 0), (0, 1, 0), (0, 0, 3)$. I need to use triple integrals to solve this, so I made a diagram and set the ...
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2answers
73 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
1
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1answer
75 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
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1answer
24 views

Reference for transformation of integrals over Lipschitz boundaries

Let $D\subseteq\mathbb{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Then according to page 314 of [Function spaces, Alois Kufner, 1977] one can define a surface integral of a real valued function ...
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2answers
112 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
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1answer
28 views

Tangent line to a curve statement

I am having problems understanding some parts of the proof of some statement related to tangent line to a curve. I'll copy the exact statement and proof and then my doubts. Statement If $\mathcal C$ ...
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0answers
73 views

Does chain rule require continuously differentiability?

Recently I read the book Advanced Calculus written by Fitzpatrick. The Theorem 15.34 tells that If $F:\mathbb R^n\to\mathbb R^m$ is CONTINUOUSLY differentiable (all partial derivatives exist and ...
3
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2answers
109 views

how to prove gradients vectors are the same in polar and cartesian coordinates.

Suppose $T=T(r,\theta)=G(x,y)$ How do you prove $\nabla T(r,\theta)=\nabla G(x,y)$? I can think of some arguments in favor of this equality, but I want an actual proof or a very good intuitive ...
2
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1answer
107 views

Transformation rule for partial derivatives

I can't fathom the step I have highlighted in green. Am I using the chain rule in 3 dimensions? What is it that I am transforming here?
2
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3answers
326 views

Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...