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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0answers
30 views

Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$

I'm trying to solve this problem: Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$ First, I founded the gradient: $\nabla f(x,y,z)= (2x\cos{z}, ...
10
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1answer
280 views

Maxima of the function $\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$

I am looking for extrema of the function $$g(a,b):=\left \vert \int_{-1}^1 e^{i(ax+bx^2)}dx \right \vert$$ where $a,b >0$ are real parameters. I already plotted this function and got the ...
4
votes
1answer
111 views

To find the volume dilation, integrate the determinant of the Jacobian

On the road toward proving the change of variables theorem in several variables, is there a painless way to show that $$\text{Vol}(\phi(U))=\int_{U}|\text{det}(d\phi)|,$$ where $\phi$ is $C^1$, ...
0
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1answer
32 views

Finding minimum value of a directional derivative(Multivariate Calculus)

Let $f(x, y) = x ^2 e^{−y^2}$ and $v = (1, 1)$. Find all points $(x, y)$ where $|Dvf(x, y)|$ has its minimum value. What i tried. I know that im order to find the mimimum value of a directional ...
0
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1answer
66 views

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ prove the following:

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ such that $ f(0)\neq -1$ and $\displaystyle\int_{0}^{b} f(t) \, dt=0$ Show that the equation $\displaystyle\int_{x}^{a} f(t) ...
1
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1answer
32 views

Differential equation multivariable solution

I don't understand how I would solve the following problem: Where does the $F(t,y) = -5$ come from? I tried solving it normally, do I create a multivariable function that satisfies $F(t,y) = -5$? ...
0
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2answers
279 views

Absolute Max/Min of a function of two variables on a set?

How do you find the absolute maximum/minimum values of the function $f(x,y) = x^2 + y^2 - 8y + 16$ on the given set R where $R = {(x,y): x^2 + y^2 ≤ 25}$ I know the absolute maximum is 81 and ...
0
votes
3answers
54 views

Show that the limit of $\frac{\sin(x) - \sin(y)}{x+y}$ does not exist.

Trying to show that $$\lim_{(x, y) \to (0, 0)}\dfrac{\sin(x) - \sin(y)}{x+y}$$ does not exist, but I'm having a lot of trouble. So far I've tried splitting the expression into two parts, but ...
0
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1answer
42 views

Line Integral with Arclength Parametrization

Suppose we have an arclength parametrization of a curve in the $xy$-plane given by $x(s)$, $y(s)$ where $0 \leq s \leq L$. We want to integrate a scalar function $f(x,y)$ along this line. Since we are ...
2
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2answers
28 views

Are the plane and the line parallel?

Problem In the picture above are given the coordinates of the points $O(0,0,0)$, $A(6,0,0)$, $C(0,12,0)$, $D(0,0,5)$, $K(0,6,5)$, $L(6,12,4)$, $M(6,8,0)$, $N(0,8,0)$. It seems as if line $KL$ is ...
0
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0answers
13 views

Is it possible for the derivative of a multivariate function to be a function of lesser dimension?

Let's say I have some function $f$ such that $f'(a,b,c,d)$ exists for all $a$, $b$, $c$, and $d$, and that $f(a,b,c,d)$ is dependent upon $a$, $b$, $c$, and $d$. (That is, $f(a,b,c,d)$ can't be ...
0
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1answer
42 views

Misconception about chain rule in multiple variables

Let $z=f(x,y)=e^{x}\sin(xy)$, $x=g(s,t)$ and $y=h(s,t)$. If $k(s,t)=f(g(s,t),h(s,t))$, find $\displaystyle\frac{\partial k}{\partial s}$. Until now, I have found that: $\displaystyle\frac{\partial ...
1
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1answer
55 views

A version of the Fundamental Theorem of Calculus for two variables

Let $f(x,y)$ be differentiable in the rectangle $R=[a,b]\times[c,d]$, show that the function $\displaystyle F(x,y)=\int_{a}^{x} f(t,y) \, dt$ is also differentiable in $R$ and that ...
1
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2answers
38 views

mulivariable calculus-distance and planes

With 4 points A B C D, how do I find the distance from point D to the plane through A, B, C? This is a rather basic calc question I know but I'm not sure where to start. I imagine I'd probably have to ...
6
votes
1answer
99 views

Smoothness in $\mathbb{R}^n$

Embarrasingly simple question, but I got the feeling that I cannot see the forrest for the trees right now: If I have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ and want to show that it is ...
0
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1answer
33 views

Compute differential of Even function

Let $f:\mathbb{R^n}\rightarrow \mathbb{R} $, If $f$ is a differentiable function and $f(-\vec{x})=f(\vec{x}) ,\forall \vec{x} \in \mathbb{R^n}$. Compute the differential of $f$ in $\vec{0}$. I don't ...
-1
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1answer
50 views

Calculate Brieskorn Manifold? [duplicate]

I need show that Brieskorn Manifold is submanifold with dimension $2n-1$ and calculate specifically for $d=2$ and $n=1$ $W(d)=\lbrace (z_{0},z_{1},...,z_{n})\in \mathbb{C}^{n+1}\vert$ $ ...
2
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0answers
42 views

Lagrange multiplier for more than one constraints.

How to minimize $x^TAx$ over the set $D=(x\geq 0, x^TBx=1$ and $(I-A^\dagger A)x=0$), where $A$ is copositive matrix of order $n-1$ and $B$ is strictly copositive matrix of order $n$. If I drop the ...
1
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0answers
32 views

Smooth function on a closed set.

Evans book on PDE's defines for a given open subset $U$ of $\mathbb{R}^{n}$, $C^{k}(\overline{U})=\lbrace u:U\rightarrow \mathbb{R}^{n}$, such that $D^{\alpha}u$ exists and is uniformly continuous ...
2
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3answers
80 views

Conplex/real Integration and poles of function

So I am working on the following problem: Let $\Delta $ be the unit disk centered at origin, and $f$ is holomorphic on $\Delta-\{0\}$. If $$\int_\Delta|f|dxdy<\infty$$ show that $f$ has at most a ...
0
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1answer
59 views

On the multivariable Taylor expansion

Apparently the second order multivariable Taylor expansion is: $$f(\mathbf x+\mathbf h)=f(\mathbf x)+ \partial_i f(\mathbf x) h_i + \frac 12 \partial_j \partial_i f(\mathbf x + t \mathbf h) h_i h_j$$ ...
0
votes
1answer
21 views

Equation of the Tangent Plane to the Surface

Find the equation of the tangent plane to the surface $$ z= \exp\Big(\frac{3x}{17}\Big)\ln(4y) $$ at the point $(4,1,2.808)$. I got $$ 0.4955512784x+0.506408305y+0.319386581, $$ but this is ...
1
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1answer
40 views

Limit of $\frac{x^2-y^2}{x^2+y^2}\sin(x-3y)$ when $(x,y) \to (0,0)$

Show that $$\lim_{(x,y)\to (0,0)}\frac{x^2-y^2}{x^2+y^2}\sin(x-3y)$$ Does not exists I've tried the traditional patches, but I always find zero as answer. Any hint? Thanks in advance!
0
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1answer
16 views

If partial derivatives of order $m$ are continuous then all partial derivatives of order $\leq m$ are also continuous

I'm working on this problem, I've tried to solve it by using the definition for partial derivative but haven't been able to prove it. Appreciate any help: Prove that if all partial derivatives of ...
0
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2answers
58 views

Limit of $\frac{xy^3}{1+y^3}$ when $(x,y) \to (0,0)$

Show that $$\lim_{(x,y)\to (0,0)} \frac{xy^3}{1+y^3} = 0$$ The only way I know of doing it is using squeeze theorem, but I couldn't find any function. Any hint? Thanks!
0
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1answer
59 views

Is a multivariable function continuous iff it is continuous with respect to each variable?

I am very uncertain when it comes to understanding the continuity of multivariable functions. If we have, for example, a function $f: \mathbb{R}^{4} \to \mathbb{R}$, and we denote the four variables ...
1
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0answers
11 views

Limit in 2 variables for $r=||(x,y)||$

Show that the function $$f(x,y)=\exp\left(\frac{1}{r^2-1}\right), if \;r<1$$ $$f(x,y)=0, if \; r \geq 1$$ Where $r =||(x,y)||$ is continuous in $R^2$ If $r>1$ or $r<1$ it is clearly ...
0
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1answer
108 views

Surface area of a sphere by cylindrical coordinates

I was resolving a problem of electromagnetism that I needed relating the 1/4 surface area of sphere with electrical field. Well, using spherical coordinates is very easy to do that. Take a look: ...
0
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1answer
73 views

Show f is continuous

$$ f(x,y)= \begin{cases} \frac{1}{x^2+y^2} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y)=(0,0) \end{cases} $$ Show that f is a continuous function
1
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1answer
36 views

How to set up an integral by these conditions?

I've got these surfaces: $$ z = 0\\ z = 4 - y^2 $$ And a cylinder: $$ x^2+y^2=4 $$ I need to find the volume enclosed by these figures. As far as I understand the limits of integration for $z$ are ...
0
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1answer
19 views

Vector variable and single-variable gradient

How do I show that $$ f(\overrightarrow{x}) = \nabla f( \overrightarrow{0} ) \cdot \overrightarrow{x} $$ for every $x$, given that $f \space (t \cdot \overrightarrow{v})= t \space f( ...
2
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0answers
14 views

Show that the innerproduct of two vector is a differentiable mapping

The question is to show that the innerproduct is a differentiable mapping. Define $g: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ such that $ g(x_1,x_2) = <x_1|x_2>$. We use the ...
1
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1answer
36 views

How do I find the limit for part c?

Find each of the following limits, or explain that the limit does not exist. Let $f(x,y) =\begin{cases} x^2&, x \geq 0\\ x^3&, x < 0 \\ \end{cases}$ a) limit of $f(x,y)$ as ...
1
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1answer
46 views

Find a normal to a surface defined by $F(u,v) =0$, $u = xy$, $v = \sqrt{x^2+z^2}$

This is a problem from Apostol, Calculus, Volume II (p. $302$, Chapter $9.8$). The three equations $F(u,v) = 0$, $u = xy$, and $v = \sqrt{x^2 + z^2}$ define a surface in $xyz$-space. Find a ...
1
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1answer
62 views

Computing the unit vector for a generalised helix

The space curve $$\mathbf x (t) = \begin{pmatrix} \cosh t \\ \sinh t \\ t \end{pmatrix}$$ is an example of a generalized helix, meaning that its tangent vector makes a constant angle $\theta$ with a ...
0
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1answer
60 views

Exam Question from multivariable calculus.

This question from a previous multivariable calculus exam.I don't know how to start with this question: Let $f$ be differentiable at every point of line segment joining $x_0$ and $x_0+h$.Show that ...
0
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1answer
31 views

Question regarding minimum

I'm wondering whether the following should be true. Suppose $f(t)$ is a real valued function (say, on $\mathbb{R}^n$) which attains its minimum at a unique point, say $x^*$ in the closure of a set ...
0
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1answer
41 views

If $u = e^{x+y} + \ln (x^3+y^3-x^2y-xy^2)$, find the value of -

If $u = e^{x+y} + \ln (x^3+y^3-x^2y-xy^2)$, find the value of $$x^2 \dfrac{\partial^2u}{\partial x^2} +2xy \dfrac{\partial^2u}{\partial x \partial y} +y^2 \dfrac{\partial^2u}{\partial y^2} + x ...
0
votes
0answers
93 views

Setting up an integral for a physics question.

The problem begins like this: a charge distribution is given by $\rho(r,\theta,\phi)=\gamma r^3cos\theta,a<r<b,0\le\theta<\pi/2$ and is zero everywhere else. The distance from the origin is ...
0
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0answers
11 views

Show that $\vec \varepsilon(t, \vec x) = \vec E e^{-i(\omega t - \vec k \cdot \vec x)}$ satisfies Maxwell's equations

Let $\vec k$ be a vector in $\Bbb R^3$ and let $\omega = |\vec k|$. Fix $\vec E \in \Bbb C^3$ with $\vec k \cdot \vec E = 0$ and $\vec k \times \vec E = i \omega \vec E$. Show that $$\vec ...
3
votes
2answers
66 views

Proving maximum of dot product using derivatives

I am curious to know whether there is a way to prove that the maximum of the dot product occurs when two vectors are parallel to each other using derivatives. In particular, given: $c = ...
2
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2answers
39 views

Can the partial derivative of f(x,y) at (a,b) exist if f(x,y) is not continuous at (a,b)?

Suppose f(x,y) is continuous for all $(x,y) \neq (a,b)$, (not continuous at (a,b)), can the partial derivative with respect to x (or y) at (a,b) still exist?
3
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1answer
104 views

Two ways to evaluate $\int (\Delta u) v d\Omega$, two different results

I would like to evaluate the integral $\int (\Delta u) v d\Omega$, where the domain $\Omega$ is a cylinder. On the boundaries, either the normal derivative $\partial_n u$ is zero or $v$ is zero. An ...
0
votes
1answer
134 views

Multivariable optimization - how to parametrize a boundary?

A metal plate has the shape of the region $x^2 + y^2 \leq 1$. The plate is heated so that the temperature at any point $(x,y)$ on it is indicated by $T(x,y) = 2x^2 + y^2 - y + 3$. Find the ...
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2answers
90 views

Understanding what is a gradient vector?

Definition : The Gradient Vector. Let A be an open set in a euclidean space $X$.Let $f:A\rightarrow \mathbb R$ be a real valued differentiable function.Then,for each $a\in A$,there is a unique ...
0
votes
0answers
30 views

Smoothness depends on parametrization

Does the smoothness (meaning infinitely differentiable) of a function depend on its parametrization? Suppose we have the function $f(t) = [t^2,t^{\frac{1}{3}}]^T$ on [0,1]. Then $\nabla f(t) = ...
3
votes
1answer
70 views

Vector Calculus intuition: Why is the magnitude of a velocity vector the speed?

From my understanding of basic Calculus (which could very well be completely flawed), the derivative of position with respect to time would give us the slope at every point of that function, which ...
4
votes
1answer
369 views

Finding continuity and differentiability of a multivariate function

Determine whether the following functions are differentiable, continuous, and whether its partial derivatives exists at point $(0,0)$: (a) $$f(x, y) = \sin x \sin(x + y) \sin(x − y)$$ ...
5
votes
1answer
68 views

Non-differentiability of a function of two variables at a point

I have problems understanding this: Function $\;g(x,y)\;$ is given, for which a) $\;g_x(0,0)=7\;$ b) $\;g(t+2t^2,\sin3t+4t^2)=5e^t\;$ c) $\;\lim_{t\to 0}\frac{g(t,2t)-g(3t,4t)}t=10\;$ They ask ...
0
votes
3answers
37 views

Limits of two variable functions

For the limit as $x,y$ go to $0$ of $\frac{xy}{\sqrt{x^2+y^2}}$, when I change the equation to polar coordinates I recieve $\frac{r\cos(\theta)r\sin(\theta)}{r}$. I then factor out and $r$ from both ...