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
0answers
24 views

Show that $\iint_S (n\times \nabla)f\, dS=\int_C tf\, ds$ and $\iint_S (n\times \nabla)\times f \,dS=\int_C t\times F\, ds$

If $S$ be a closed region lying on a surface and bounded by the curve $C$ and $n$ be the unit positive normal vector to $S$, and $f$ and $F$ be two fields with continuous fiest derivatives in $S$, ...
3
votes
2answers
53 views

Prove that $\int_S n\times r dS=0$

If $r$ be the position vector of a point on a closed surface $S$ and $n$ be the unit normal (outward) vector to $S$, then prove that $$\int_S n\times r\,dS=0$$ Attempt: $r=xi+yj+zk$, ...
0
votes
0answers
27 views

A problem of vector integration: Show that $\iint_S f grad f \times dS =0$

For any scalar field $f$, show that $\iint_S f\, \nabla f \times dS =0$. I don't have an idea to solve. Please help me.
2
votes
1answer
36 views

Is level set near a maximum value a circle?

Let $f$ be a $C^2$ function defined on $[0,1] \times [0,1]$. Let $0 \leq f(x) < 1$ on $[0,1] \times [0,1] \setminus \left(\frac{1}{2},\frac{1}{2}\right)$ and $f(\frac{1}{2},\frac{1}{2})=1$. It has ...
1
vote
1answer
31 views

Proving that the norm of $f'(y)$ is attained at $\pm\frac{\nabla f(y)}{\|\nabla f(y)\|}$.

Consider a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$ and a point $y\in \mathbb{R}^n$ such that $\nabla f(y)\neq 0$. Prove that there exists an unit vector $x_0\in\mathbb{R}^n$ such that ...
4
votes
3answers
112 views

Proof of this definite integral?

Saw this sometime in my calculus book, from the Putnam Math Challenges listed: $$\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1 }{ \underbrace{\dots}_{n-3 \, times} \int _{ 0 }^{ ...
-1
votes
1answer
28 views

Proving some statements about the function [closed]

Let's consider the following function: $$f:\mathbb{R}^2\to\mathbb{R}:(x,y)\mapsto\begin{cases} \dfrac{xy}{x^2+y^2}&\text{if }x^2+y^2>0\\0&\text{if } x=y=0\end{cases}$$ How do I prove the ...
-1
votes
3answers
49 views

How to prove that the following function is continuous on $\mathbb{R^2}$

$$f(x,y)=\begin{cases} \dfrac{x^2y}{x^2+y^2}\Leftrightarrow x^2+y^2\not=0\\0\Leftrightarrow x=y=0\end{cases}$$
0
votes
1answer
14 views

Transversals Related to Circles and Spheres

I was wondering if anyone could provide insight into whether the intersection of 3 circles (no interior) in R^2 intersecting at a single point would be transversal. My struggle in understanding some ...
1
vote
3answers
32 views

Limit of several variables

What would be the limit of the given multi-variable functions? $$1. \lim_{(x,y) \to (0,0)} x \cdot sin \left(\frac{1}{y}\right)$$ $$2. \lim_{(x,y) \to (0,1)} \frac{e^{x}-y}{xy}$$ I tried to solve ...
1
vote
1answer
42 views

Show that this is indeed a differentiable manifold with boundary.

I want to show that the cylinder: $$C = \{ (x,y,z)\in\mathbb{R}^3: x^2 + y^2 = 1, 0 \le z \le 1 \}$$ is indeed a a differentiable manifold with boundary, this means the following: A subset $M ...
1
vote
2answers
53 views

Multivariable limit which should be simple !

How to calculate the following limit WITHOUT using spherical coordinates? $$ \lim _{(x,y,z)\to (0,0,0) } \frac{x^3+y^3+z^3}{x^2+y^2+z^2} $$ ? Thanks in advance
0
votes
3answers
55 views

Finding the area of an ellipse

Using Green's Theorem, find the area of the ellipse $\frac{x^2}9+\frac{y^2}{16}=1$. My work so far: Green's Theorem states that $\iint_R \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial ...
0
votes
1answer
36 views

Computing the Limit of Multivariable function

How do we compute the limit of following function? $$\lim_{(x,y,z) \to (0,0,0)} \frac{sin(x^2+y^2+z^2)}{x^2+y^2+z^2}$$ If someone can give me some hint then that would be great. Thanks.
1
vote
2answers
24 views

Evaluating limit of multi variables

The question: $$\lim_{(x,y)\rightarrow(0,0)}\frac{\sin(x+y)}{x+y}.$$ I tried this using $y=x$ path and $y=0$ path and both approach to the same value, $1$. My problem is to show that this value really ...
1
vote
0answers
22 views

Finding points on a surface which contain tangent planes parallel to the axis-planes

Suppose that $f(x,y,z) = \frac{3}{2-x} + \frac{1}{y-z}$ and let $S$ be the surface given by the equation $ f(x,y,z) = 1 $ Are there any points on $S$ where the tangent plane to $S$ is parallel to the ...
2
votes
1answer
33 views

Integrate a partial derivative

If we define the operator $\mathcal{G}f(t,x)= \frac{\partial f}{\partial t}(t,x)$, what is the value of $$ \int_0^t \mathcal{G}f(s, b(s)) ds? $$ I'm sure it's some subtlety in the Fundamental Theorem ...
4
votes
1answer
49 views

$\omega$ is $1$-form on $S^1$.

Let $h: \mathbb{R} \to S^1$ be $h(t) = (\cos t, \sin t)$. How do I show that if $\omega$ is any $1$-form on $S^1$, then$$\int_{S^1} \omega = \int_0^{2\pi} h^*\omega?$$
0
votes
0answers
18 views

Multivariable calculus integration over a rectangle

Please help me. I have been stuck on this question for quite a while A parallelogram S in the xy-plane has vertices $(0,0)$, $(2, 10)$, $(3, 17)$, and $(1, 7)$. (a) Find a linear transformation $u = ...
0
votes
1answer
27 views

Simple limit in multi variable II

For $x=(x_1,x_2,x_3)$ calculate (if it exists) the limit $$\lim_{x\to 0} \frac{e^{|x|^2}-1}{|x|^2+x_1^2x_2+x_2^2x_3+x_3^2x_1}$$ Solution: Let ...
2
votes
1answer
35 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
0
votes
1answer
26 views

Interchanging total derivative and partial derivative

Say I have a function $F(x,y)$, where $x = f(t)$ and $y = g(t)$. $$\frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial F}{\partial x} \tag{1}$$ $$\frac{\partial }{\partial x} \frac{\mathrm{d} ...
0
votes
1answer
15 views

Proof: Indepence of path of integration for line integral of second type

Given the line integral of the second type: $$ I = \int_{(1,\pi)}^{(2,\pi)}\left (1-\frac{y^2}{x^2}\cos{\frac{y}{x}}\right )dx + \left (\sin{\frac{y}{x}} +\frac{y}{x}\cos{\frac{y}{x}}\right )dy $$ ...
0
votes
1answer
23 views

Application of Implicit Function Thm

Problem Let $f_{1},f_{2}$: $R^{2}\rightarrow R$ of class $C^2$. Consider the zero sets $Z_{1}, Z_{2}$ (of $f_{1},f_{2}$ respectively) ie $Z_{i}=\{(x,y) | f_{i}(x,y)=0\}$. Assume $\nabla f_{i}(x,y) ...
0
votes
0answers
9 views

Dealing with gradient of generative adversarial network

I am currently working on a recurrent implementation of something called a "Generative Adversarial Network". (link: http://arxiv.org/abs/1406.2661 ) Simply explained these are two neural networks, ...
1
vote
1answer
35 views

Poincare: Change of form to a primitive 1-form

Given a 2-form w on $R^3$, such that dw=0; how does one find all 1-forms k such that dk=w? Provided a homotopy H(t) one can pull back w and apply the Poincare operator on it. But what if one does not ...
-2
votes
1answer
58 views

Symbolic contour integral evaluation

Can anyone help with the evaluation of the following contour integral : $$\oint\limits_C \phi(x,y)\,dx+\psi(x,y)\,dy.$$ Where the contour $C$ is given by: What I am looking for is how to split ...
1
vote
1answer
26 views

Laplace Operator in $3D$

I am looking to find the radial part of Laplace's operator in three dimensions. I looked up Laplace's operator in spherical coordinates and from there I guess the radial part is: ...
2
votes
1answer
54 views

Proof of Hamilton's equation from integral invariant

This is from pages 273 - 274 0f Whittaker's book of analytical dynamics. Its in the public domain. Let $q_1,q_2,\ldots,q_N$ be functions of time. And let $p_1,p_2,\ldots,p_N$ also be functions of ...
1
vote
0answers
25 views

Laplacian of composition

Let $U \subset \mathbb{R}^n$ be open and $u \in C^2(U)$ with $\Delta u(y)=0$ for all $y \in U.$ Let $\phi: V \rightarrow U$ be in $C^2(U)$, too with $V \subset \mathbb{R}^n$ open. Now, I want to ...
1
vote
2answers
37 views

Solving a triple integral with Spherical Coordinates

I am attempting to solve this integral $\iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2)}} \mathrm{dV}$ where the region $v$ is a unit sphere. How would I go about converting the function inside the integral ...
1
vote
2answers
64 views

Why the set $g^{-1}(\{0\}) $ is not a differentiable manifold?

Let $g:\mathbb{R}^2 \to \mathbb{R}$ given by $g(x,y) = x^2 - y^2$. Then I am triying to figure out why this function is not a differentiable manifold , I was trying to give an explicit coordinate ...
3
votes
1answer
25 views

How can I show that this function is smooth?

I got an assignment which I just can't find the right way to solve and I hope that someone could help me out here. It goes like this: Let $\Omega\in R^n$ be a domain and $b_1,...,b_n:\Omega\to R^n$ ...
3
votes
1answer
36 views

Simple limit in multi variable

For $x=(x_1,x_2,x_3)$, determine the limit $$\lim_{x\to 0} \frac{\sin|x|^2}{|x|^2+x_1x_2x_3}. $$ I want to use that $\lim_{x\to 0} \frac{sin|x|^2}{|x|^2} = 1$ but I can't see how to do that. Any ...
0
votes
0answers
21 views

Newton-Raphson in multiple variables, function not equal to 0

I'm currently solving a nonlinear system of equations using a jacobian matrix consisting of differential quotes - however the equation itself isn't equal to zero. At present, the function looks like ...
0
votes
1answer
17 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 ...
0
votes
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
vote
0answers
50 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
votes
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
votes
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 ...
0
votes
0answers
22 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
votes
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
vote
1answer
27 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
vote
0answers
10 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
votes
4answers
91 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
votes
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
vote
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
votes
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
vote
0answers
12 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 $. ...
0
votes
1answer
30 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.