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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1answer
19 views

Set up triple integral for volume (cylindrical coordinates)

I am given the following question Let $D$ be the region in $\mathbb{R}^3$ that lies within $x^2 + y^2 =4$, underneath the surface $z= 4- x^2 - y^2$ and above the surface $z=- \sqrt{9-x^2 - y^2}$ ...
1
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2answers
26 views

Finding the Limits of the Triple Integral (Spherical Coordinates)

Let $D$ be the region in $\mathbb{R}^3$ below $z=-\sqrt{x^2 + y^2}$ and above $z=-\sqrt{4-x^2 -y^2}$. Rewrite \begin{align*}\iiint \limits_D z^2 dV\end{align*} using Spherical Coordinates. I ...
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2answers
17 views

Maximum rate of change along which curve?

The temperature $T(x,y)$ at points in the $xy$-plane is given by $T(x,y)= x^2 -2 y^2$. An ant wishes to cool off as quickly as possible. Along what curve through $(2,1)$ should the ant move in order ...
1
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1answer
25 views

General change of coordinates

I would like to integrate over the following surface: $\Omega=\{(v_1,\dots,v_n):\sum_{i=1}^N\phi(|v_i|^2)=N, \sum_{i=1}^N v_i=p, v_i\in R^3,p \in R^3\}$. If $\phi(|v|^2)=|v|^2$, it is easy to see ...
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0answers
44 views

a simple calculation

Can anyone see how (59) lead to (60)? Here $$ \Phi_{\varepsilon}(s):=(s+\varepsilon)^k-{\varepsilon}^k $$ and $\varepsilon$ is just a small number. $b\in C(\bar{Q}_T)$, where $Q_T=\Omega\times ...
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0answers
19 views

Calculate D(f o g)(1,2)

I'm doing this problem: Let: $g:\mathbb{R}^{2} \rightarrow \mathbb{R}^2$ and $f:\mathbb{R}^{2} \rightarrow \mathbb{R}^2$ be a differentiable function such that: $g(0,0)=(1, 2); \ \ g(1,2)=(3,5); \ \ ...
-1
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0answers
10 views

Find Parametric equations for the tangent line to the curve [on hold]

Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 + y2 and the ellipsoid 7x2 + 5y2 + 2z2 = 20 at the point (-1, 1, 2). (Enter your answer in terms of ...
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3answers
22 views

Prove that $d_n$ is a Cauchy sequence in $\mathbb{R}$

Let $(x_n$) and $(y_n)$ be Cauchy sequences in $\mathbb{R}^n$ , i.e. lim$_{n,m}$ |$x_n$ − $x_m$| = $0$ and lim$_{n,m}$ |$y_n$ − $y_m$| = $0$. For each n, let $d_n = |x_n − y_n|$. Prove that $d_n$ is a ...
0
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1answer
16 views

Graphing the derivative at a point

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function. The function gives the surface $\{x,y,f(x,y)\}$ in $\mathbb{R}^3$. Fix a point $(a,b)\in\mathbb{R}^2$. The derivative of $f$ at $(a,b)$ is the ...
1
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1answer
20 views

Intuition behind divergence?

$\overrightarrow f = 3x\overrightarrow i - 3y\overrightarrow j$ $\overrightarrow g = 3x\overrightarrow i + 3y\overrightarrow j$ If I calculate the divergence of $f$ I get $0$. If I calculate the ...
1
vote
1answer
15 views

If $f$ is $C^1(U)$) , are $D_i f_j$ where $i=1,\ldots,n$ and $j=1,\ldots,m$ are all continuous on $U$?

$f$ is a function from an open set $U$ in $R^n$ to $R^m$ then $f=(f_1,f_2,\ldots,f_m)$, I am confused whether the following are true: If $f$ is continuous on $U$, does that imply that ...
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0answers
20 views

Question concerning what a certain notation means. Its usually in calculus

$x$ln$(1 + \frac{1}{x}) = 1 +$ln$(1 + \displaystyle\sum_{i=1}^n \displaystyle\frac{a_{i}}{x^i}) + O(x^{-n-1})$ for $x \rightarrow \infty$ and $n \in \mathbb{N}$. My question is as follows:I have seen ...
0
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0answers
27 views

function differentiable but not continuously differentiable

Can I found an example of a function $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ such that $f$ differentiable but it is not continuously differentiable and it is not invertible on a point??? Any idea? ...
0
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1answer
46 views

Show that this function is convex?

So I'm supposed to show that this function is convex, but I have no idea how to go about it...I've been told to use Cauchy Schwarz in order to show that the Hessian is non-negative definite, but I'm ...
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2answers
57 views

Show that the subsets of the plane are open

I know that this set is open but I just don't know how to prove it. $A = \{(x,y) | -1< x <1, -1< y <1\}$ I looked at all the problems in my book but it only had problems like $B = ...
4
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1answer
34 views

Is a sphere really a (differentiable) manifold?

I am a beginning student in Differential Geometry. According to what I understand, the charts: $$\sigma_+^z(x,y) = (x,y, \sqrt{1 - x^2 - y^2} )$$ $$\sigma_+^x(u,v) = (\sqrt{1 - u^2 - v^2},u,v )$$ ...
2
votes
1answer
50 views

Prove that g is continuous.

Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a continuous function and define $g : \mathbb{R}^{n} \rightarrow \mathbb{R}$ by $g(x) = |f(x)|, x \in \mathbb{R}^{n}$ Prove that g is ...
1
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1answer
54 views

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

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) + ...
2
votes
1answer
30 views

Jacobian matrix of the inverse of a bijective function

Let $f:\mathbb{C}^n\rightarrow\mathbb{C}^n$ be a function such that $f=f(f_1,\ldots,f_n)$ and $f_i=f_i(x_1,\ldots,x_n)$. Also, $f$ is bijective and its Jacobian matrix exists. Does$f^{-1}\,$Jacobian ...
-2
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0answers
28 views

Gauss curvature K in polar coordinates

EDIT: A surface is given in Monge's form: $z=f (x,y)$ the partial derivatives of $z$ are.. $$ p = \frac{\partial z}{\partial x}, \; q = \frac{\partial z}{\partial y}, \; ...
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2answers
28 views

$n$-th order derivatives of multivariable composites

Suppose I have smooth functions $g: \mathbb{R} \to \mathbb{R}^3$, $f: \mathbb{R}^3 \to \mathbb{R}$. Is there a nice expression for the $n$-th derivative of $f \circ g: \mathbb{R} \to \mathbb{R}$ in ...
1
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2answers
24 views

Finding the scalar component of $\overrightarrow{PQ}$ in the direction of $\overrightarrow{PR}$

Context I am given the 3 points: $P(3,-1,3)$, $Q(1,-1,6)$, and $R(5,0,1)$ I know that $\overrightarrow{PQ} =(3-1)\hat{i}+((-1)-(-1))\hat{j}+(3-6)\hat{k} $ $=2\hat{i}+0\hat{j}-3\hat{k}$ and ...
0
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0answers
25 views

Solution to Laplace's Equation in different coordinate systems

Find the general solution to Laplace's equation for spherical symmetry (everything can only depend on $r$, the radius), cylindrical symmetry (everything can only depend on $s$, the radius), and planar ...
0
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2answers
36 views

Determine if the following set is open, closed or neither {$x\in\mathbb{R}^{2} | x_1 + x_2 = 1$} $\subset \mathbb{R}^{2}$

my textbook doesn't have solutions to most these problems, and this one is really giving me some trouble. Any help is appreciated. Determine if the following set is open, closed or neither ...
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0answers
15 views

Integral of harmonic function in a ball

Let $f\in C^2(\Omega)$ an harmonic function in $\Omega$, and: $$ \phi(r) = \frac{1}{2\alpha_2r} \int_{\partial B_r(x)} f(y) d \sigma(y) $$ Prove that $\phi '(r)=0$ by calculating the line integral. ...
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0answers
9 views

Notation for expansion of multivariable functions

Let $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ be some analytic function. I want to say something like, as $ x, y \rightarrow 0 $, the taylor expansion looks like: $$ f(x,y) = a x^2 + b y^2 + ...
0
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2answers
58 views

Necessary and Sufficient Conditions for $f_{xy} = f_{yx}$

When beginning to study multivariate calculus, you almost immediately come across the Schwarz-Clairaut Theorem which gives sufficient conditions to guarantee that mixed partials are equal. For the ...
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2answers
13 views

what is asking to do to show that u(x,t)=$\phi(x)\theta(t)$ is a solution to the wave equation?

I been asked the following if $\phi^{''}(x)+a^2 \phi(x) =0$ and $\theta^{''}(t)+c^2a^2 \theta (t) =0$ show that $u(x,t)=\phi(x)\theta(t)$ is a solution to the wave equation My question is do I have ...
1
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2answers
38 views

Taking the partial derivative of $z=F(x/y)$

I have just started on chapter for partial derivative and I have a very basic question: Please go easy on me since I just started on the topic today. Let the function be: ...
0
votes
1answer
24 views

Find local max/min and saddle points of $f(x,y) = e^x\cos(y)$.

I want to find the local max/min and saddle points of $f(x,y) = e^x\cos(y)$. I started off by finding the following: \begin{align} f_x &= e^x\cos(y) \\ f_{xx} &= e^x\cos(y) \\ f_y &= ...
1
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1answer
33 views

Multivar limit $\frac{6x-2y}{9x^2-y^2}$ by approach

I'm resolving the limit of $\frac{6x-2y}{9x^2-y^2}$ when $(x,y)\to(1,3)$ We didn't study any special theorem, we did only approach. I tried first the changes $y=mx$ and $y=x^2$. In $f(x,mx)$ ...
0
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1answer
26 views

Multivar limit $\frac{(2x^2).y}{x^4+y^2}$

I'm resolving the limit of $\frac{(2x^2).y}{x^4+y^2}$ when $(x,y)\to(0,0)$ We didn't study any special theorem, we did only approach. I tried first the changes $y=x^2$ and $y=0$. In $f(x,x^2)$ ...
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3answers
25 views

Computing $\lim_{(x,y)\to(0,0)}(x^2+1)\cdot\frac{\sin y}{y}$

Hi I have a limit with two variables in front of me and the book says directly that it is equal with $1$ but for the life of me I dont understand why?? maybe the answer is stupid but I am excausted ...
1
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0answers
22 views

Are two definitions of differential function equivalent?

Definition 1: From https://www.math.hmc.edu/calculus/tutorials/tangentplanes/differentiability.pdf or Differentiability for a function of two variables A function $f(x, y)$ is differentiable at the ...
1
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0answers
50 views

$f(x,y)=\frac{x^ay^b}{x^2+y^2}$ differentiable?

One knows that $$f(x,y)=\frac{x^\alpha y^\beta}{x^2+y^2}$$ is continuous iff $\alpha+\beta>2$. Is there any condition of $\alpha$ and $\beta$ so that the previous function is differentiable?
2
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1answer
58 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
27 views

does the following function have all directional derivatives?

$$xy\sin(\frac{1}{xy})$$ the function has partial derivatives at every point , but i wanted to know whether this function had directional derivatives at every point? for $x=0$ the function is ...
0
votes
1answer
54 views

Does $f$ have a local maximum?

Let $f:\mathbb{R}^{3}\rightarrow{\mathbb{R}}$, $f\in{C^{2}}$, can I claim, that if $f$ have a critical point $\widehat{x_{0}}$ and is such that every eigenvalue of $D(\nabla(f))$ is positive then $f$ ...
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0answers
21 views

I want explain to me proof of Chain rule in multi-Variable functions simple?

I want explain to me proof of Chain rule in multivariable functions simple?
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0answers
21 views

Calculus III chain rule and parametrics

Parametrize the upper half of the unit circle by x =cos(t), y =sin(t), for 0<=t=>pi. Let T = f(x, y) be the temperature at the point (x,y) on the upper half circle. Suppose that dT/dx = 6 x - 3 y ...
0
votes
3answers
65 views

Prove that $\lim \limits_{(x,y,z) \to (0,0,0)} \frac{{xyz}}{{x+y+z}}=0$

I have a strong feeling that the following limit is zero, can anybody help me prove it. $ \lim\limits_{(x,y,z) \to (0,0,0)} \frac{{xyz}}{{x+y+z}}$ Thanks!
2
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1answer
22 views

Sketch parametric curve

An exercise in my textbook asks to sketch the parametrical curve of the following equation: $$x=e^t\cdot\cos(t)\\y=e^t\cdot\sin(t)\\t\ge0$$ I would usually try to solve one of the equations for t and ...
1
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1answer
22 views

Find max rate of change of $f(x,y) = \sin(xy)$ at the point $(1,0)$ and in the direction in which it occurs.

Find max rate of change of $f(x,y) = \sin(xy)$ at the point $(1,0)$ and in the direction in which it occurs. I did the following: $$\nabla f = <\frac{\partial f}{\partial x}, \frac{\partial ...
1
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0answers
10 views

Find $\frac{dz}{dt}$ if $z(x,y) = x^2y + xy^2$; $x(t) = 2 + t^4$; $y(t) = 1 - t^3$.

Find $\frac{dz}{dt}$ if $z(x,y) = x^2y + xy^2$; $x(t) = 2 + t^4$; $y(t) = 1 - t^3$. I used the Chain Rule: $\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial ...
0
votes
2answers
26 views

Minimum of a function $f(x,y)=\frac{(1+2y)(1+\frac{x}{2})}{(1+y)(1+x)+x}$

what is the minimum of a function \begin{align} f(x,y)&=\frac{(1+2y)(1+\frac{x}{2})}{(1+y)(1+x)+x}\\ \text {s.t. }& 1 \le y \le x \le y(1+y) \end{align} I asked Wolfram and Alfa and it says ...
0
votes
2answers
52 views

$\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ . what can you say about the direction of $\vec{b} \times \vec{c}$?

I know that $\vec{a} \times \vec{b}$ and $\vec{c} \times \vec{d}$ are perpendicular therefore the dot product would equal $0$.
1
vote
1answer
31 views

Volume of a solid bounded by surfaces - is it correct?

Could you check if my calculations and reasoning are correct. And maybe suggest a nicer way of solving this problem? We are given a solid bounded by these surfaces: $y=x^2, \ y=1, \ 2x+y+z = 4, \ ...
2
votes
0answers
19 views

Continuity and Differentability of a Partial derivative

Pick the correct statement (a) If $f(x,y)$ is continuous everywhere, then it is also differentiable everywhere. (b) If $f(x,y)$ is continuous everywhere, then it also has partial derivatives ...
1
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0answers
29 views

3D graph of a function - its projection onto xy plane?

Suppose we have a function with two variables $x, \ y$, ie $f(x,y)=z$ Is there any universal, general way of determining the projection of the graph of this function, after intersecting it with a ...
1
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0answers
11 views

Homework-Second derivative of a multivariate function

Let $f(x):\mathbb{R}^n\rightarrow\mathbb{R}$, and let $\theta(\alpha)=f(x+\alpha s)$ where $\alpha\in\mathbb{R}$ and $x,s\in\mathbb{R}^n$, the goal is to find $\theta''(\alpha)$. I guess that the ...