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 (1)

5
votes
1answer
1k views

Is any divergence-free curl-free vector field necessarily constant?

I'm wondering, for no particular reason: are there differentiable vector-valued functions $\vec{f}(\vec{x})$ in three dimensions, other than the constant function $\vec{f}(\vec{x}) = \vec{C}$, that ...
4
votes
4answers
139 views

Convolution integral $\int_0^t \cos(t-s)\sin(s)\ ds$

How can I calculate the following integral? $$\int_0^t \cos(t-s)\sin(s)\ ds$$ I can't get the integral by any substitutions, maybe it is easy but I can't get it.
4
votes
2answers
151 views

$\frac{d}{dx}(b^TAx)$ where $b, x \in R^{n\times 1}$ and $A \in R^{n\times n}$

How do you differentiate the following expressions with respect to the vector $x$. I think I might be a little conceptually confused on what happens when you take the derivative with respect to a ...
3
votes
1answer
77 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = (-y^3,x^3,z^3)$ - 2012 9C

Question: 2012 9C. Consider the (cutoff) paraboloid defined by $z= x^2 + y^2 , \frac{1}{9} \le z \le 1$. Sketch the surface. Verify Stokes’s Theorem for for $\mathbf{F} = (-y^3,x^3,z^3)$. Herein, I ...
3
votes
1answer
580 views

l'Hôpital's Rule and Multivariable Limits

I decided to post another message regarding this problem because I still didn't understand it at all: Can someone give me an example of function $f(x,y),g(x,y)$ for which: $\lim\limits_{r\to 0^+} ...
3
votes
2answers
156 views

Proof of: If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$.

Let $f: \mathbb R ^n\to\mathbb R$ be a differentiable function. If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$. Where can I find a proof for this theorem? ...
3
votes
1answer
206 views

Let $f(x,y)=\frac{x^3-y^3}{x^2+y^2}$. Is f differentiable in $(0,0)$?

Let $$f(x,y)=\frac{x^3-y^3}{x^2+y^2}$$ My solution manual says that this function is not diffb. in $(0,0)$ because it is not linear. Well my problem is that I don't see why this function is linear, ...
3
votes
1answer
116 views

Differentiability in $\mathbb{R}^{2}$

Here is my question: Find all the points $\left ( x,y \right )$ in $\mathbb{R}^{2}$ where the following function is differentiable: $f\left ( x,y \right )=\left | e^{x}-e^{y} \right |.\left ( x+y-2 ...
3
votes
2answers
376 views

Total Derivative and Multilinear Functions

So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review: Given a function ...
2
votes
0answers
58 views

Difficult multivarate random variable - how to calculate it?

I have a random variable defined by $Y=\frac{\sum_{j=1}^{N}l_j \cos\theta_j}{\sum_{j=1}^{N}l_j\sin\theta_j}$ where $l_j \sim \text{log-normal-distribution} (\left \langle l \right \rangle, \sigma _l)$ ...
2
votes
2answers
81 views

How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole

This seems pretty trivial but I'm not sure what to do. My coordinates are Cartesian, and I want to send point the $(x_1,...,x_{n-1},x_n)$ to point $(0,...,0,-1)$ so that all the other points are also ...
2
votes
1answer
133 views

Lagrange multiplier constrain critical point

When using Lagrange multipliers in an inequelity, $$ f(x,y) = x^2+y $$ with the constraint $$ x^2+y^2 \leq 1. $$ I have to find the critical points inside the "disk" right? I've done $$ f_x = 2x ...
2
votes
2answers
181 views

Other way to express $e^{|x|+|y|}$

I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
2
votes
1answer
234 views

Calculate the volume between $z=x^2+y^2$ and $z=2ax+2by$

I'm trying to calculate the volume between the surfaces $z=x^2+y^2$ and $z=2ax+2by$ where $a>0,b>0$. Here's what I've tried: First I noticed the projection of the volume to the xy plane is a ...
2
votes
2answers
1k views

A continuously differentiable map is locally Lipschitz

Let $f:\mathbb R^d \to \mathbb R^m$ be a map of class $C^1$. That is, $f$ is continuous and its derivative exists and is also continuous. Why is $f$ locally Lipschitz? Remark Such $f$ will not be ...
2
votes
1answer
358 views

Find the Jacobian

Find the Jacobian $$\frac{\partial(x,y)}{\partial(u,v)}$$ for $x=u^2+v^2$, $y=u^2-v^2$. My solution: I tried solving it as it is by using the Jacobian matrix (determinant?) and got my answer to be ...
2
votes
2answers
1k views

What is the vector form of Taylor's Theorem?

I checked most of the posts about Taylor expansion with scalar functions. Could anyone tell me what is the multivariate version of Taylor's Theorem, and how I can use it?
1
vote
2answers
63 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 ...
1
vote
1answer
88 views

Express partial derivatives of second order (and the Laplacian) in polar coordinates

$z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$ Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$ I'm having big troubles with using chain rule, in particularly the second ...
1
vote
1answer
73 views

Compute the flux of $(z \sin x, yz \cos x, x^2 + y^2)$ through the paraboloid. [closed]

Given the vector field $$F(x, y, z) = \langle z \sin x, yz \cos x, x^2 + y^2 \rangle,$$ calculate the flux $\int_S F \cdot \hat{n} \; dS$ through the paraboloid $$S = \{(x,y,z) : z = -3(x^2 + y^2) + ...
1
vote
1answer
74 views

Continuity conditions for multivariate functions.

Is the following true ? A proof or counter-example or reference would be nice. A function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is continous at $(0,0)$ if and only if if for all $a, b$, the limits ...
1
vote
1answer
58 views

Solve Multivariate Polynomial

Is there a general way to solve a multivariate polynomial (example here: http://mathworld.wolfram.com/MultivariatePolynomial.html) Say for instance I knew some function $F(x,y) = xy + x^2 + y^3 + ...
1
vote
2answers
97 views

How to make sense of this calculus notation, Advanced College Level

I have $f(x)$=$(2x,e^x)$ what does this notation mean? Notation: $Df(\frac{∂}{∂x})$ Certainly $Df(x)$=$(2,e^x)$ but how can I replace $x$ with $\frac{∂}{∂x}$? Particularly, how can I make sense of ...
1
vote
1answer
137 views

Evaluate the line segment intergal

Evaluate the line integral $$\int_C xe^{y}\, {\rm d}s,$$ where $C$ is the line segment from $(-1,2)$ to $(1,1)$. I do not get this part of calculus at all please show me how this is solved and if ...
1
vote
1answer
271 views

shortest distance between two points on $S^2$

Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$ Length of a curve in $3D$ is ...
1
vote
1answer
79 views

Question Concerning Vectors

I am given the information that $ \vec{u} = \langle 1, 1/2 \rangle$ and $\vec{v} = \langle 2,3 \rangle$. There are a few pieces I am asked to find, and these are the one I am having trouble with: ...
1
vote
2answers
201 views

Do we need additional assumptions for problem 3-37 (b) in Spivak´s calculus on Manifolds?

Problem 3-37 (b) reads: Let $A_{n}=[1-1/2^{n},1-1/2^{n+1}]$. Suppose that $f:(0,1)\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for any $x\notin$ any $A_{n}$. Show that ...
1
vote
1answer
951 views

What is the meaning of evaluating the divergence at a _point_?

Reading this first, Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative ...
1
vote
1answer
171 views

Hölder continuity of a function from $[0,1]$ to $[0,1]^2$

I'm trying to prove that if $g: [0,1] \longrightarrow [0,1]^2$ is an $\alpha$-Hölder continuous mapping whose image is the entire square $[0, 1]^2$ then $\alpha \leq 1/2$. I wouldn't know where ...
1
vote
0answers
148 views

Help to show this equality of differential operators in spherical coordinates

There are two points on the same sphere with coordinates ${R, \theta_1, \phi_1}$ and ${R, \theta_2, \phi_2}$. Also I have the operator $\displaystyle { \nabla _{{\Omega _1}}^2 + \nabla _{{\Omega ...
1
vote
1answer
221 views

passing the Derivative inside an integral

Question: Suppose we have: $F(x)=\int_{a(x)}^{b(x)}e^{h(x,t)}dt$. Is it true that $F^{'}(x)=\int_{a(x)}^{b(x)}\frac{\partial h(x,t)}{\partial x}.e^{h(x,t)}dt$ ? Please tell me under what ...
0
votes
2answers
39 views

Second Order Partial Differentiation

I don't have a clue on how to start this question. I have a feeling I will need to use the Clairaut's theorem: $f_xy=f_yx$ Can anyone advise?
0
votes
2answers
87 views

2 examples to try to understand partials derivatives and deriviability

To prove that a functions has partial derivatives every partial has to exist, and every partial exist only if the limit of definition of partial exist. Is this right? Then if partials exist ,and the ...
0
votes
0answers
48 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
0
votes
1answer
74 views

Triple Integral Volume Question

The question asks for the triple integral of $e^{-(x^2+y^2+z^2)^{3/2}} dV$ where $D$ is a sphere of radius $4$. The answer that I came up with is $2(1-e^{-64})$. However, I am not confident in this ...
0
votes
0answers
28 views

Lower boundary for hessian eigen values

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
0
votes
1answer
118 views

Can anyone help me with these double Integrals using mathematica

$$ \int_0^6 \int_0^4 \frac{\sqrt{(1+x^2+y^2 )^2+4 ( x^2+y^2)}}{1+x^2+y^2}\, dy \, dx$$ And $$\int_{-1}^1 \int_{-y}^y \frac{1}{(1+y^2)^2} \sqrt{(1+y^2)^4 + 4x^2(1+y^2)^2 + 4y^2(1+x^2)^2} \, dx \, dy$$ ...
0
votes
1answer
42 views

Multivariate integration of a derivative w.r.t. a single variable

$x=(x_{1},...,x_{n})$. If $\frac{\partial g(x)}{\partial x_{l}}=f(x_{l})$ for $l=1,...,n$, should we have $g(x)=\sum_{l=1}^{n}\int f(x_{l})dx_{l}+c$? If yes, what's the theorem or proposition behind ...
0
votes
1answer
66 views

Multivariable Calculus, Two Path Test.

I'm have trouble understanding how you determine which paths you should choose. In the book for the function lim as (x,y) goes to (0,0) $$f(x,y) = \frac{2x^2y}{x^4+y^2}$$ They say "We examine the ...
0
votes
3answers
424 views

Show discontinuity of $\frac{xy}{x^2+y^2}$

How to show this function's discontinuity? $ f(n) = \left\{ \begin{array}{l l} \frac{xy}{x^2+y^2} & \quad , \quad(x,y)\neq(0,0)\\ 0 & \quad , \quad(x,y)=(0,0) \end{array} ...
0
votes
3answers
80 views

When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$

When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$, where $f_{xy}$ and $f_{yx}$ denote the mixed (second) partial derivatives of a multivariable function $z=f(x,y)$.
0
votes
1answer
100 views

Explanation on a proof of a property of mollifiers

Here are some definitions that was taken of PDE Evans book: Here is a proof of a property of mollifiers: My (elementary) question is: Why is the convergence uniform on $V$? Thanks.
0
votes
3answers
580 views

How does one prove that if function's partial derivative respect to every variable is zero, function is constant?

How does one prove that if function's partial derivative respect to every variable that the function defines over is zero function is constant function? I just noticed it, but I cannot prove it.
0
votes
3answers
215 views

Green theorem of curve

Use Green's Theorem to evaluate the integral $$\int\limits_C \left(y-x\right) \mathrm dx+\left(2x-y\right) \mathrm dy$$ for the path C defined as $x=2\cos\theta \;\text{and}\; y=2\sin\theta.$ Here is ...
0
votes
2answers
95 views

Determining the Moment of inertia

Let $a,b,c$ be positive real numbers such that $c<a$. Suppose given is a thin plate $R$ in the plane bounded by $$\frac{x}{a}+\frac{y}{b}=1, \frac{x}{c}+\frac{y}{b}=1, y=0$$ and such that the ...
0
votes
1answer
303 views

Green's Theorem

Let $C$ be the closed,piecewise curve figured by traveling in straight lines between the points $(-2,1),(-2,-3),(1,-1),(1,5)$ and back to $(-2,1)$, in that order. Use Green's Theorem to evaluate the ...
0
votes
1answer
340 views

Computing $\iint \sin(4x^2 +2y^2)\, \mathrm dA$ over an elliptical region

$$\iint \sin(4x^2 +2y^2)dA,$$ where the region is bounded by ellipse $4x^2 + 2y^2 = \pi$ and the lines $y = 0$ and $y = \sqrt{2}x$. This looks like a change of variables integral. Need some hints. ...
0
votes
1answer
205 views

Multivariable Calculus, Jacobian

In the xy-plane, draw the region R bounded by the lines $y = 1+x, \quad y= -1 +x, \quad y= -1-x, \quad y= 1-x$ Use double integral in rectangular and polar coordinates to find the area of R you ...
0
votes
1answer
146 views

Arc length of level sets

I have a function $z = B \sin x \ \sin y+\cos x \ \cos y$. Where $0 \leq x \leq \pi$ and $0 \leq y \leq \pi$. I need to find the length of the curve that describes a level set for any value of $B$. ...
0
votes
1answer
551 views

Finding a perpendicular line that connects two skew 3D lines at a particular distance?

I have two linear, skew, 3D lines, and I was wondering how I could find a point on each of the lines whereby the distance between the two points are a particular distance apart, and that the vector ...