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).

learn more… | top users | synonyms

0
votes
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!
2
votes
1answer
104 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} = ...
0
votes
2answers
53 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
votes
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
votes
2answers
78 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
votes
1answer
169 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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
3answers
323 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 ...
1
vote
1answer
43 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 ...
1
vote
1answer
73 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 . ...
1
vote
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 $f(x, y)$ ...
2
votes
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 ...
3
votes
2answers
265 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) ...
0
votes
2answers
89 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 ...
0
votes
2answers
132 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 ...
0
votes
0answers
15 views

trace of hyperbola 2 sheets

for yz parallel trace book mentions there is no trace unless |x|>|a| then the trace is an ellipse it is hard for me to visualize this any insight would be appreciated, why is this so? sorry I know ...
-1
votes
2answers
70 views

Convergence of $\int (1+|x|)^{-\alpha} dx$

I want to prove that $$\int_{\mathbb R^n} (1+|x|)^{-\alpha} dx < \infty$$ if and only if $\alpha > n$, but I have no idea how to generally prove this. It is easy to see for $n=1, 2 \ldots$ but I ...
1
vote
1answer
79 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( ...
1
vote
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,\; ...
2
votes
3answers
105 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 ...
0
votes
1answer
172 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 ...
1
vote
1answer
65 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 ...
1
vote
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 ...
0
votes
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: ...
0
votes
1answer
22 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 ...
1
vote
2answers
56 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 ...
8
votes
1answer
485 views

taylor's formula in $\mathbb{R}^n$ as exponential of derivative operator

I hope this question is not too vague. There is some relationship between the formal operator $$e^\frac{d}{dx} = 1 + \frac{d}{dx} + \frac{1}{2!}\frac{d^2}{dx^2} + \ldots $$ and taylor's formula. ...
1
vote
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 ...
2
votes
1answer
33 views

Parametrizing to Calculate Flux

Evaluate the flux of $\mathbf{f}$ across the oriented surface $\Sigma$ by computing the surface integral $\iint_{\Sigma} \mathbf{f} \cdot d\sigma$, where $\Sigma$ is the surface $z=xe^y$ for $0 \leq x ...
1
vote
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 $\mathcal C$ ...
4
votes
2answers
111 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 ...
3
votes
2answers
99 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 ...
3
votes
0answers
72 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 ...
1
vote
2answers
55 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
2
votes
1answer
97 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
votes
2answers
42 views

Confusion about Spherical Coordinates Transformation

We have a function $$f(x,y,z) = \frac{e^{-x^2 -y^2 -z^2}}{\sqrt{x^2+y^2+z^2}}$$ and we want to integrate it over the whole $\mathbb{R}^3$. Then what i got is the following: $$\int_{\mathbb{R}^3}^ \! ...
1
vote
1answer
29 views

Evaluate $\int_C \nabla(r^4) \cdot \hat n ds$ in terms of moments of inertia…

Curve $C$ is closed plane curve, $a$ and $b$ are the moments of inertia about $x$ and $y$ axes, $\hat n$ is the unit outward vector and $r = \left|x \hat i + y \hat j\right|$. Here's what I have: ...
2
votes
1answer
442 views

Rankine Hugoniot Jump Condition Derivation

I follow the majority of this derivation I just do not understand where the two terms highlighted in green come from.
2
votes
1answer
100 views

How to find the second derivative of an implicit function?

We know from multivariable calculus that if $y(x)$ is a function given implicitly by the equation $F(x,y) = 0$, then $$ \frac{dy}{dx} = -\frac{F_x}{F_y} \tag{1} $$ This is quickly proved by applying ...
0
votes
1answer
28 views

Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
0
votes
1answer
171 views

How to find an equation of the plane, given its normal vector and a point on the plane? [duplicate]

I have a question regarding vectors: Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
2
votes
1answer
55 views

Area of a Curved Surface

Find the area of the part o the surface $z=xy$ that lies within the cylinder $x^2+y^2=1$. I'm not sure how to set up the surface integral to compute this.
0
votes
2answers
32 views

First order approximation of multivariable function

Let $O$ be an open subset of $\mathbb R^2$ and suppose the function $f:O\to\mathbb R$ is continuous at the point $(x_0,y_0)$ in $O$. Define tangent plane as $\phi(x,y)=a+b(x-x_0)+c(y-y_0)$ where ...
0
votes
1answer
58 views

Help with double integral

I need to prove if this integral exist (and some others) but i would like to know if there is a condition to say if the integral exist (for example in this case) that would help me solve this kind of ...
0
votes
1answer
52 views

Triple integral visualisation problem with a sphere and a cylinder

Write a triple integral in cylindrical coordinates for the volume of the solid cut from a ball of radius 2 by a cylinder of radius 1, one of whose rulings is a diameter of the ball. I am unable to ...
1
vote
2answers
49 views

A curious question about optimizing a function of 2 variables.

Let $f(x,y)$ be defined and has continuous first and second partials on a domain $D$. Also, let $$A = \frac{\partial^2 f}{\partial x^2} \\ B = \frac{\partial^2{f}}{\partial x \partial y} \\ C = ...
2
votes
1answer
30 views

If $\vec r = x \hat i + y \hat j + z \hat k$ and $r =| \vec r |$ show that $curl [f(r) \vec r] = 0$

I know that $\nabla \times f(r) \vec r = \nabla f(r) \times \vec r + f(r) \left ( \nabla \times \vec r \right )$. I figured that the rightmost expression is $0$. How do I prove that $\nabla f(r) ...
2
votes
1answer
68 views

Differentiability at a point

Let $f:\mathbb{R}^{2}\mapsto\mathbb{R}\mathbb{}^{2}$ be given by $$f(x,y) = \left(\begin{array}{c} x^{2}y+2y-x\\ 3xy+4y \end{array}\right)$$ Find a open set containing (0,0) where f has a ...
0
votes
1answer
56 views

Proving or disproving continuity of a function

Consider a function $f:\mathbb{R}^{n}\times \mathbb{R}^{+} \rightarrow \mathbb{R}$, with the property that for a fixed vector $a:=(a_1,a_2,\cdots,a_n) \in \mathbb{R}^{n}$, there exist a finite ...