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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0
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1answer
30 views

Word Problem Lagrange Method

I am studying for my exams and got very very stuck at a word problem on the Lagrange Methods, my biggest difficulty is to properly identify the function to be maximized (in this case) and so its ...
1
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2answers
51 views

Local extrema and minima of the multivariable function $f(x,y) = x^2y+y^2+xy$

Let $f(x,y) = x^2y+y^2+xy$ be a function, I want to find its local extrema an minima. I easily find that $f$ has 2 critical points: $(x,y)=(0,0)$ and $(x,y) = (-1,0)$. In order to find its local ...
0
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1answer
19 views

Trying to find the square with minimum area inscribed in a Square of side L

A square has side length of L. Using the lagrange's multipliers, show that all squares inscribed in the square of side length of L, the square with minimum area has a side length of (sqrt(2)/2)L. I ...
2
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2answers
28 views

$\lim_{(x,y)\to (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2}$

I must admit that I've forgotten how to do multivariable limits. Nevertheless I need to know whether the following exists: $$\lim_{(x,y)\to (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2}$$ Would it be as ...
3
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2answers
19 views

Integrating Line Segment and Path

Just wondering if anyone can give some assistance. I'm stuck on an exam question: $\ \vec F (x,y,z) = (6xy + 4xz)\vec i + (3x^2 + 2yz)\vec j + (2x^2 + y^2)\vec k , x,y,z ∈\Bbb R.$ Evaluate $$\ ...
2
votes
1answer
53 views

$\iint_{\mathbb R^2}\sqrt{\frac{x^2}{a^2}+\frac{x^2}{b^2}}e^{-\frac{x^2}{a^2}+\frac{y^2}{b^2}}dxdy$

$$\iint_{\mathbb R^2}\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}\,e^{-\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)}\,dx\,dy$$ Basically I have done problems similar to this, using the theorem that if ...
0
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3answers
22 views

How to approach $\phi: \Bbb R^3 \to \Bbb R$ such that $\vec F = \nabla\phi.$

I'm looking through exam papers and I'm lost on what to do when asked to find a function $\phi: \Bbb R^3 \to \Bbb R $ such that $ \vec F = \nabla\phi.$ An example of a question I'm looking at is as ...
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1answer
10 views

Finding the Components of a Hessian Matrix of a Quadratic Form

I'm trying to find the Hessian form of the following quadratic form: $f(x,y) = x^2y+y^2+xy$. I know that it's in the form of a matrix and that the elements of $H_f(a)_{i,j}=\dfrac{\delta^2f}{\delta ...
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0answers
27 views

Intersection of a smooth plane curve and a circle

Let $\gamma(t)=(x(t),y(t)):[0,2\pi] \rightarrow \mathbb{C}$ be a $C^1$-Jordan curve. How to show that there is a small circle $C$ that intersects $\gamma$ only at two points?
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2answers
45 views

consider a function $(x,y)$ given by… [duplicate]

Consider the following function given by; $f(x,y) = \begin{cases} \frac{xy}{x^2+y^2},&(x,y) \neq 0 \\\\ \ \ \ \ 0,&(x,y)=0 \end{cases}$ Is the function differentiable in the ordinate pair ...
-5
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0answers
34 views

Find $\oint_C x^2y\,dy+x\,dy.$ [closed]

MultiVariable Calculus Question ( about green's Theorem and etc.) 5) \begin{align*} C_1 &: \vec r(t) = \langle t, 0\rangle&0\leq t\leq 1\\ C_2 &: \vec r(t) = \langle1, ...
0
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1answer
22 views

Order of variables when computing the Jacobian for the purposes of calculating the change of variables factor?

Consider the transformation that converts from polar to cartesian coordinates: \begin{align} x &= r\cos{\theta} \\ y &= r\sin{\theta} \end{align} To compute the change of coordinates, we ...
0
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1answer
26 views

Multivariate chain rule

I am applying the steepest descent algorithm with an exact line search for a particular choice of $f$. If $\mathbf x$ and $\mathbf p$ are fixed vectors, and I define $g \colon [0,\infty) \to \mathbb ...
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1answer
15 views

Volume bounded between to multivariable functions

Really would like some pointers on how to attack this! I understand how to find the integrand, but how do you get the bounds?. The correct answer is Choice B.
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0answers
28 views

Finding the area of a circle that is formed by cutting a sphere.

Say I have a sphere $x^2+y^2+z^2=a^2$ and a plane $x+y+z=b.$ How do I find the surface area of this surface? I think I would use the surface integral and for graphs the surface "element" is ...
3
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0answers
82 views
+50

Intuitively what is the second directional derivative?

I'm thinking that the second directional derivative, if both dd's are evaluated in the same direction, will just give you the concavity (the second scalar derivative) in that direction. Is that ...
1
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2answers
35 views

How is f(4,4,4)=48 a local minimum? Can it be inferred that it is either a maxima or minima & only one extreme value within the constraint?

Disclaimer: In the definition (Stewart Calculus, 7E): "Method of Lagrange Multipliers" part (b)- Evaluate $f$ at all extreme points $(x,y,z)$ from step a. The largest of these values is the maximum ...
2
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0answers
33 views

Spherical, polar coordinates, volume of set.

Find the volume: $$\{(x,y,z)\mid x^2+y^2 \leq (z-1)^2 \leq 4-\frac{x^2}{2} - 2y^2, z\geq 1 \}$$ I've got the intersection of the following two basically: \begin{align} 1. & & & (z-1)^2 ...
1
vote
1answer
33 views

Local Immersion Theorem in $\mathbb{R}^n$ proof

I am trying to prove the following: Let $U \subset \mathbb{R}^n$ be open and $f \in C^1(U;\mathbb{R}^m)$. Let $x^\star \in U, \ y^\star = f(x^\star)$. Suppose that $\mathrm{d}f(x^\star)$ ...
0
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2answers
40 views

An approach to proving that continuous partial derivatives implies differentiable

I had an idea for how to prove that if $f$ has continuous partial derivatives, then it's differentiable. To make things simpler, take a two variable function $f(x, y):\mathbb R^2 \to \mathbb R$. Let's ...
2
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1answer
22 views

Contraction Banach theorem

Given the following function: $$g(z)=C*\begin{pmatrix} x^2+y^2-2 \\ x^2-y^2-1 \\ \end{pmatrix}+z, \; \; \;z=(x,y)\in [0.93,1.52]\times [0.41,1]$$ Prove that $g ...
0
votes
1answer
11 views

infinte dimensional nonlinear optimization

Is there any structured way to tackle $$\min_{\{x_n\},\{y_n\}}\frac{(\sum_{n=1}^{\infty}x_ny_n)^2}{\sum_{n=1}^{\infty}y_n^2\sum_{n=1}^{\infty}x_n^2}$$ when $x_n, y_n \neq 0$, ...
0
votes
2answers
16 views

About the interpretation of line integrals

I've been asked to compute the line integral of the function $f(x,y)=xy$ over the elipse $\frac{x^2}{4}+y^2=1$ counterclockwise orientated. My doubt is if this means that i have to compute the surface ...
0
votes
4answers
29 views

Can the directional derivative be thought of as a rate of change of function with respect to some arbitrary paramter on which all its inputs depend?

The formal definition of a directional derivative is $$\nabla_{\vec{v}}f(\vec{a}) = \lim_{h \to 0}\frac{f(\vec{a} + h\vec{v}) - f(\vec{a})}{h |\vec{v}|}$$ Let's take a function with constant x and ...
2
votes
1answer
32 views

Derivative formalism question

After seeing a lot of integrals going like $$\int f(x,y) \,dxdy = \int f(x,y)\,dA$$ I am wondering wether it is allowed to write something like this: $$\frac{d f(x,y)}{dA} = \frac{\partial^2 ...
1
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2answers
31 views

Convergence of integral power of $\cos$

Find the integral and find $k$ in order to converge ($k$ is real number). $$\int_0^{\frac{\pi}{2}} \cos (\theta) ^{2k} d\theta.$$ I can find the value of integral if $k$ is integer, but what happens ...
1
vote
1answer
14 views

Is $h(x,y) = g(x)f(y)$ convex, if $g(x)$ and $f(y)$ are convex in $x$ and $y$, respectively?

I want to check if $h(x,y) = g(x).f(y)$ is convex, given that g(x) is convex and decreasing in $x$ on its and $f(y)$ is increasing and convex in $y$. Is there any way to check/prove the convexity of ...
0
votes
0answers
26 views

Surface integral of $\int\int z\ dS $ where $S$ is the surface $z = x^2 + y^2$, $x^2 + y^2 \le 1$

The question asks the following: Evaluate $\int\int z\ dS $ where $S$ is the surface $z = x^2 + y^2$, $x^2 + y^2 \le 1$. I'm just looking for feedback on whether I have done this correctly - my ...
0
votes
1answer
31 views

Space curve. Stokes' theorem

I was trying to apply Stokes over the toroidal spiral with equation, for example, $$x = (4 + \sin(20 t))\cos(t),\ y = (4 + \sin(20t))\sin(t),\ z = \cos(20t).$$ I'd like to know if somebody has any ...
0
votes
2answers
31 views

Evaluating the integral of $f(x,y,z) = \frac{y}{\sqrt{z}}$ on $y \geq 0$ and $0 \leq z \leq x^2$ and $(x-2)^2+y^2 \leq 4$

I am asked to evaluate the integral of $f(x,y,z) = \frac{y}{\sqrt{z}}$ on $$ y \geq 0\\ 0 \leq z \leq x^2\\ (x-2)^2+y^2 \leq 4 $$ What I have so far (and it seems a little off) is $$ ...
0
votes
1answer
17 views

centroid of a right triangle

I'm asked to find the $M_x, M_y$ and the centroid of the shape created by the functions $5x/6$ and $x=6$ that has a density of $5$. I find $$M_y \int_0^5 \frac{5}{6} x^2dx = \left. \frac{5}{18} x^3 ...
2
votes
0answers
76 views

Expectation or Integration of the normal cdf

Can any one help me how to solve this pronbelm? I have a random variable $W$, i.e., $$W=\Phi(X)^k\Phi(-X)^m=P(Z\le X)^kP(Z \ge X)^m,$$ $X$ is Normal($\mu$,1), $Z \text{ is Normal(0,1)}$, and $k$ ...
0
votes
2answers
42 views

Is the Hessian symetric for $z^3+y^2+xy+yz+3x-3z$?

I wish to study the function $f(x,y,z) = z^3+y^2+xy+yz+3x-3z$ and find its extreme values. I search for values for which $\nabla f(x,y,z) =0$. $$\frac{\partial f}{\partial x} = y+3$$ ...
0
votes
1answer
23 views

Brackets and Vector Calculus Operations

Is this operation: $$ \nabla\cdot\nabla u $$ The same as this operation: $$ (\nabla \cdot \nabla)u $$ The same as this operation: $$ \nabla \cdot (\nabla u) $$
2
votes
2answers
67 views

League of Legends optimal items

In the popular game League of Legends, your effective amount of hit points ($E$) against physical damage is a function of your actual hit points ($H$) and the amount of armor ($A$) you have. $$E = ...
0
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1answer
34 views

Proving an equality using the Leibniz integral rule.

If $f:[0,b]\times [0,b]\to \mathbb{R}$ is continuous, proof $\int_{0}^{b}\mathrm{d}x\left(\int_{0}^{x}f(x,y)\, \mathrm{d} y\right) = \int_{0}^{b}\mathrm{d}y\left(\int_{y}^{b}f(x,y) \,\mathrm{d} ...
0
votes
0answers
16 views

Integration by parts and a vector field

The solution to my problem should be just integration by parts, but I can't see it. We have an integral of the form \begin{equation} \int_Uf(y)e^{ikg(y)}dy \end{equation} where $U\subset \mathbb R^n$ ...
2
votes
1answer
81 views

Help evaluate $\int_0^\infty x\operatorname{erfc}(a + b\ln (x)) \,dx$.

I am trying to evaluate $$ I = \int_0^\infty x\operatorname{erfc}(a + b\ln (x)) \,dx $$ where $a \ge 0$ and $b> 0$. $$ I = \frac{2}{\sqrt{\pi}}\int_0^\infty \int_{a + b\ln (x)}^{\infty} ...
0
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0answers
15 views

Lefschetz Fixed Points and Invertible Derivative Matrices

Background: Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be continuously differentiable. Def: $x\in\mathbb{R}^n$ is a fixed point if $f(x)=x$ Def: $x\in\mathbb{R}^n$ is a Lefschetz fixed point if ...
0
votes
3answers
34 views

One partial derivative is continuous (at a single point) implies differentiable?

Let $f:\mathbb{R}^2\to\mathbb{R}$ and $(p,q)\in\mathbb{R}^2$ such that both $f_x$ and $f_y$ exists at $(p,q)$. Assume that $f_x$ is continuous at $(p,q)$. How do we prove/disprove that $f$ is ...
0
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0answers
19 views

Definition of higher order Fréchet derivative

My lecturer has recommended to us that we check that the obvious two candidates for the $k^{th}$ order Fréchet derivative are the same. That is defining the $k^{th}$ order Fréchet derivative ...
4
votes
2answers
74 views

How to solve simultaneous inequalities?

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1 ≤ 2x+y ≤ 2$$ $$0 ≤ ...
0
votes
0answers
17 views

The average value of f on region R (double integrals)

If I am given f(x,y) and I calculate the average of the region given by the formula in this image should the average always = (min of f(x,y) +max of f(x,y))/2. I think this is only the case if the ...
3
votes
3answers
550 views

How do I solve a double integral that has an absolute value?

First off, I apologize for any English mistakes. I've come across a double integral problem that I haven't been able to solve: Find $\int_0^\pi \int_0^\pi \left |\cos(x+y) \right| \,dx\,dy$ ...
1
vote
1answer
28 views

Problem with integration limits using spherical substitution

Good night, i have a problem with this integral, please help me with the integration limits. \begin{align} ...
1
vote
1answer
17 views

Problem with integration limits with cylindric cordinates.

Good night, i have a problem when i go to verify the integration limit $0\leq\theta\leq\varPi/2$ because i think the integration limit go to $0\leq\theta\leq\varPi$ because is an half a circle. ...
2
votes
1answer
59 views

Product of two uniform random variables/ expectation of the products

Suppose I want the expectation, $E\Phi(X-\mu)\Phi(\mu-X)$, where $\Phi(.)$ represents the Normal CDF, and X is $Normal(\beta,1)$. Consequently $\Phi(.)$'s are uniform[0,1] and at the same time two ...
1
vote
1answer
37 views

Problem solving an multiple integral with integration limits.

Good night, i have a serious problem solving this integral. $\int_{0}^{2}\int_{0}^{\sqrt{2x-x^{2}}}\int_{0}^{a}z\sqrt{x^{2}+y^{2}}dzdydx$ I make a change of cylindrical coordinates, and when i make ...
0
votes
1answer
25 views

Bounding a gradient of a function

Defining an infinitely differentiable fucntion $\phi$ as $$\phi(x) = \left(\frac d2 \right)^{-n} \int_{\mathbb R^n} \psi\left( \frac{y-x}{d/2} \right) \, dy,$$ I need to show that $$|\text{grad} \, ...
1
vote
0answers
35 views

Finding the Maximum and Minimum values w/constraint [duplicate]

I apologize I have asked this question before but it died and I just got around to working it out based on the suggestions so here it is. Let the function $f$ be defined as $f$($x$,$y$,$z$) $=$ ...