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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2
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1answer
21 views

Partial derivative w.r.t. to the time of a time-dependend quadratic form

Suppose the quadratic form $$V(x(t), t) = \frac{1}{2} x^\mathsf{T}(t) P(t) x(t)$$ where $$x(t) \in \mathbb{R}^n,~P(t) \in \mathbb{R}^{n \times n},~\text{and}~P(t) = P^\mathsf{T}(t) > 0$$ ...
1
vote
1answer
39 views

quick question on surface integrals/stokes theorem

Say if I have a cylinder with the bottom removed I have an open surface. When we apply stokes theorem (or carry out the surface integral) are we just summing over the outer surface or the surface ...
1
vote
2answers
39 views

Multivariable taylor polynomial

$$f(x, y) = e^{2x+xy+y^2}$$ Find the 2nd order taylor polynomial to the above function about (0,0) The formula is: $$P(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)+\frac ...
2
votes
2answers
70 views

evaluate the double integral $\int_0^4 \int_{\sqrt{y}}^2 \sqrt{x^2+y}\, dxdy$

evaluate the double integral $\int_0^4 \int_{\sqrt{y}}^2 \sqrt{x^2+y}\, dxdy$ Hi all, could someone give me a hint on this question? I've actually tried converting to polar coordinates but i cant ...
1
vote
1answer
28 views

Surface integral (algebraic solution)

Find the area of the part of the cone $$z^2=x^2+y^2$$ that lies inside the cylinder $$x^2+y^2=2ay$$ I would like an algebraic solution. This is how I set it up: $$\int\int_Sds = ...
0
votes
1answer
36 views

Equality form of second order Taylor series

I am reading a book on optimization wherein a statement using Taylor's expansion is made without proof. \begin{equation} f(\mathbf{y}) = f(\mathbf{x}) + (\mathbf{y} - \mathbf{x})^T\nabla ...
1
vote
1answer
95 views

Definition of a 2-variable function derivative

I read this definition in a book of multivariable calculus: $f(x,y)$ is differentiable at $(x_0,y_0)$ if it can be expressed as the form $$f(x_0+\Delta x, y_0+\Delta y)=f(x_0,y_0)+A\Delta ...
1
vote
1answer
48 views

The number of vertices in a polytope is finite [duplicate]

I want to prove the following: Let $K$ be a convex polytope. Show that $K$ has a finite number of extreme points. I have seen the bound for the cardinality of the set of extreme points: $|E| \leq ...
1
vote
2answers
48 views

Multivariable limit exists?

Does the limit $$\lim_{(x,y)\rightarrow (0,0)} \frac{y^4}{x^\beta(x^2+y^4)}$$ exists for $\beta>0$? I don't think it exists but how do you prove it rigorously. Thanks
1
vote
2answers
206 views

How to get projection of ellipsoid onto sphere

I'm trying to get the projection of an ellipsoid onto a sphere. Depicted in the image below, I need the projection of the red ellipsoid onto the unit sphere at the origin. I have tried various ...
0
votes
1answer
68 views

definite integral of piecewise function in R2

For $t\ge 0$ let $$ f(x,t) = \begin{cases} x, & \text{if $0\le x \le \sqrt{t}$} \\ -x+2\sqrt t, & \text{if $\sqrt t \le x \le 2\sqrt t$} \\ 0, & \text{otherwise} \end{cases} $$ ...
1
vote
0answers
63 views

Definite Integral of $\sqrt{(x^2+y^2)^k+B}$

I'm trying to evaluate the integral $$ \int_{-1}^1 \sqrt{(x^2+y^2)^k+B} \, \mathrm{d}y $$ WolframAlpha doesn't return a response even for simplified versions of this, but I believe it can be ...
0
votes
1answer
48 views

Deriving the Maclaurin series of $\int_{0}^{x} \frac{e^t+e^{-t}-2}{t^2}dt$

I've found this question to be a bit tricky: $\int_{0}^{x} \frac{e^t+e^{-t}-2}{t^2}dt$. My first thought, knowing that $e^t = \sum_{n=0}^{\infty} \frac{t^n}{n!}$, is to break the equation into three ...
0
votes
1answer
82 views

Differentiability implies continuity in $R^2$

Let F be a function from $R^2$ to $R^2$. F is differentiable at a point (a,b) in $R^2$, prove that F is continuous at this point. Can i write F(x,y)= F(a,b)+ c(x-a)+ d(x-b)+e where c,d,e are real ...
0
votes
1answer
31 views

Double integral with parametrization

Evaluate $$\int\int_D(-2x^2+2xy+1)\,dx\,dy $$ where $$D: x^2+y^2≤a^2$$ I have parametrized as follows: $$x=a\cos t$$ $$y=a\sin t$$ $$\int_0 \int_0^a(-2a^2cos^2t+2a^2\cos t\sin t+1)r\,dr\,dt $$ ...
0
votes
1answer
31 views

Show that the sum of the oscilations is less or equal to $f(b)-f(a)$

I want to show the following: Let $ f:[a,b]\to \mathbb{R}$ be an increasing function.If $ x_1,\ldots,x_k\in[a,b]$ are different, show that $$\displaystyle\sum_{i=1}^k O(f,x_i) < f(b) - f(a).$$ ...
0
votes
1answer
105 views

Find the point on the paraboloid $z = \frac{x^2}{4}+ \frac{y^2}{25}$ that is closest to the point $(3, 0, 0)$

Find the point on the paraboloid $z = \frac{x^2}{4}+ \frac{y^2}{25}$ that is closest to the point $(3, 0, 0)$ Hi all, could someone give me a hint on how to start doing the above question?
0
votes
2answers
149 views

Finding slope at a point in a direction on a 3d surface

This is not a duplicate, I have attempted the question and am not sure why my answer is incorrect. QUESTION: The surface with equation $z = x^3 +xy^2 $ intersects the plane with equation $ 2x−2y = 1$ ...
2
votes
1answer
116 views

Showing differentiability for a multivariable piecewise function

Let $$f(x,y)=\begin{cases} xy\sin(x/y) & y\neq 0 \\ 0 & y=0\end{cases},$$ show whether $f(x,y)$ is differentiable at $(0,0)$. It seems that there are multiple ways to do this but ...
0
votes
0answers
48 views

The formula of Eclidean distance to a hyperplane.

I have a hyperplane eqution H: "$X - Y = 0$" where $X, Y \in R^{n\times m}$. Could you tell me how to deduce the smallest Euclidean distance formula for any point ($X_0,Y_0$) to H ?
1
vote
1answer
32 views

Surface integral problem

Find $$\int_S yds$$ where $S$ is the part of the plane $$z=1+y$$ that lies inside the cone $$z=\sqrt{2(x^2+y^2)}$$ What I tried to do: I combine the two equations to get the intersected surface ...
0
votes
0answers
66 views

integral of the characteristic function of a set of lebesgue measure zero

Let $A\subset \mathbb R^n$ closed rectangle and let $D\subset A$ bounded such that $D$ has lebesgue measure zero. If $\int_A \chi_D$ exists (riemann integral) then $\int_A \chi_D=0$ (where $\chi_D$ is ...
1
vote
1answer
66 views

Limit of integral in the wrong variable

Evaluate and justify: $\lim_{y\to 0^+} \int_0^1 \frac{x\cos{y}}{\sqrt[3]{1-x+y}}dx$ Can I just apply the limit to the $y$ directly, since in regards to the variable being integrated it's ...
1
vote
0answers
54 views

Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points ...
0
votes
1answer
147 views

Subharmonic function equivalent non-negative laplacian

I want to ask for a proof that if $v(x,y)$ is $C^2$ and is subharmonic [here, define as satisfyingthen $\Delta v \geq 0$ where $\Delta v = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 ...
0
votes
1answer
73 views

Continuity of a function on the complement of a set of Jordan measure zero

Let $f:D\subset \mathbb R^n \to \mathbb R$ $f= \begin{cases} c, & \text{if $\vec x \in \Omega$} \\[2ex] 0, & \text{if $\vec x \notin \Omega$} \end{cases}$ where D is a closed rectangle and ...
1
vote
0answers
22 views

parametric integral thought of as change of variable

In the video below https://www.youtube.com/watch?v=IZ5AF5mITnA At 3:20 he says, $$\int_c g(x,y)ds = \int_a ^b g(x(t),y(t)) \sqrt{x'(t)^2 + y'(t)^2}dt$$ where we can think of $\sqrt{x'(t)^2 + ...
0
votes
0answers
31 views

Find normal random variables from independent standard normal variables with correlation matrix

I'm trying to find three independent standard normal variables from three normal random variables using a correlation matrix. So far, I have decomposed the problem using the matrix's cholesky ...
0
votes
1answer
13 views

Parametrize a surface using cylindrical coord.

Hi! I am having trouble parametrizing this tower. Specifically the radius which has to be a function of the height $Z$ $$0<z<H, 0 ≤ r ≤ R(2 − z/H), \quad 0<\theta<2\pi$$ I do not ...
0
votes
0answers
18 views

from probability density function (pdf) to the probability of one event

I use a kernel density estimator for multivariate data (5 to 8 dimensions) on data in a database (numerical computations of a theoretical model, all those points provide a good estimate of a weak ...
0
votes
1answer
71 views

Find the volume V of the solid bounded by the cylinder $x^2 +y^2 = 1$, the xy-plane and the plane $x + z = 1 $.

Find the volume V of the solid bounded by the cylinder $x^2 +y^2 = 1$, the xy-plane and the plane $x + z = 1 $. Hi all, i cant seem to get the correct answer for this question. The answer is $\pi$ ...
1
vote
0answers
29 views

Continuity of partial derivative notation

Given a function $f(x,y):X \times Y \rightarrow \mathbb{R}$ where $X\subset \mathbb{R}^3$ and $Y \subset \mathbb{R}^3$. Let say I want to impose additional regularity. Assume: $f(x,y) \in ...
0
votes
1answer
47 views

Work done in a vector field

Say a particle is moving along a path $\gamma$ in a vector field, then the total work done by the force $\vec{F}$ on the particle is $\displaystyle \int_{\gamma}{\vec F}.d\vec{r}$. Say if this value ...
2
votes
2answers
47 views

Determining existence of limit with multiple variables

Given the following limit: $$ \lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3} $$ And the instrucion to "Determine whether the limit exists, give a complete argument", would the following be a "complete ...
0
votes
1answer
23 views

Derivative of a multivariable function

I have a fairly basic question that has perplexed me for a few hours now. I am trying to evaluate the derivative of a function $g(t):\mathbb{R}\rightarrow\mathbb{R}$ defined as \begin{equation} g(t) ...
0
votes
1answer
48 views

Change of variables in double integral

The problem: $\int\int_A x\ dxdy$, where A is the area delimited by the parabolas $y = x^2, y = x^2 + 4, y = (x-1)^2, y = (x-1)^2 + 4$. I figured this should be easy but somehow I'm stuck. What I ...
0
votes
1answer
232 views

Volume under hyperbolic paraboloid, above unit disk

I am asked to find the volume below $z = x^2 - y^2$ and above $x^2+y^2 \leq 1$ in the $xy$-plane. My attempt goes as follows: Since $|x| \geq |y|$ for z to be positive, that means I am looking at the ...
1
vote
0answers
49 views

Inconclusive second derivative test

I am struggling with this problem: Let $f:\mathbb{R}^n \to \mathbb{R}^m$ be $C^3$ and the second derivative $d^2$$f(y)$ = 0 at a critical point y but one of the third partial derivatives (may be ...
2
votes
2answers
99 views

crossed second partial dervatives of $\frac{xy^3}{x^2+y^2}$

Let $\displaystyle f(x,y)=\frac{xy^3}{x^2+y^2}$ if $(x,y)\neq (0,0)$, and $f(0,0)=0$. An exercise asks to compute $\displaystyle\partial^2_{xy}f(0,0)=\frac{\partial^2f}{\partial x\partial y }(0,0)$ ...
3
votes
4answers
338 views

What is the gradient with respect to a vector $\mathbf x$?

What is the meaning of "gradient with respect to $\mathbf x$"? http://en.wikipedia.org/wiki/Gradient I am talking about the symbol $$\nabla_\mathbf x$$ Does that simply mean derivative with respect ...
0
votes
0answers
138 views

Multivariable: Continuity of Piecewise function

I have this Multivariable problem.... Where I have to find out if a function is continuous or not. Here is the problem: $f(x, y)=\left\{\begin{matrix} \frac{x^4+3y^4}{x^2+y^2} & (x,y)\neq (0, ...
0
votes
0answers
149 views

Find the average height of a pyramid

Problem Description For starters can someone explain to me what the average height of a pyramid is? Does that have to do with its geometric center? The first thing I did was to calculate the height ...
2
votes
0answers
80 views

Show a multivariable function is differentiable at a point

Define $F(x,y)=(x^2, xy + y^2)$. How do I show that $F(x,y)$ is differentiable at the point $A=(a,b)$ without using any known theorems? Am I correct by showing that the limit exists, i.e. $lim_{h ...
0
votes
0answers
67 views

is F differentiable at x0,y0

$F(x,y)=(x^2-y,xy)$, how to show that F is differentiable at $(x_0,y_0)$ and find $dF(x_0,y_0)$ My attempt: I think I understand part 2 of this question, $dF=\begin{pmatrix} 2x & -1 \\ y & x ...
0
votes
1answer
38 views

Equality of mixed directional derivatives

I am trying to prove the following: Let $f:\mathbb{R}^n \to \mathbb{R}^m$ and let $f$ be $C^2$ for any vectors $u,v$. Prove that $dudv\ f = dvdu\ f$ where $du$ and $dv$ are directional derivatives. ...
3
votes
2answers
91 views

Prove that the limit doesn't exist

I have to prove that $$\lim_{(x,y)->(0,0)} \frac{xy}{2x-y}$$ doesn't exist. I have tried to use these restrictions: $x=0; y=0; y=x; y=mx; y=mx+q; y=ax^2;y=ax^2+bx+c; y=1/x, y=1/x^2,..$ and ...
0
votes
1answer
180 views

The integration of radially symmetric function

Suppose $u\in C^\infty(\overline{B(0,1)})$ is radially symmetric. i.e., there exists a function $v$: $\mathbb R^+\to\mathbb R$ such that $u(x)=v(|x|)$. Here we take $B(0,1)\subset \mathbb R^2$. ...
2
votes
1answer
75 views

Proving infinity limit of a multivariable function

I have a function $f(x,y,z): \mathbb{R}^3\rightarrow \mathbb{R}$ which is $f(x,y,z)=3x^2+z^2+y^2-2xy+14$. I'm trying to show that $f(x,y,z)\rightarrow \infty$ when $||(x,y,z)|| \rightarrow \infty$. ...
0
votes
1answer
21 views

How would I go about differentiating this (vector function)?

I want to find the gradient of this potential function \begin{align*} \phi(\mathbf{r}) = \frac{1}{|\mathbf{r} - \mathbf{r_0}|^2}. \end{align*} First, I wrote it as \begin{align*} \phi(x,y,z) = ...
0
votes
2answers
26 views

what is the bound when integrating f(x.y)=1 when x=y?

Function f is defined f(x.y)=1 if x=y and 0 if x is not equal to 0 x is [0.1] and y is [0.1] when double-integrating this, what would be the bound for integral?