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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2answers
57 views

Can I argue that $g'$ is non zero in this case?

Consider two smooth maps $g,f$ given by $$ {\partial \over \partial x} g(x)= g'(x) = \int_0^1 {\partial \over \partial x} f'(u + t(x-u)) dt = \int_0^1 f''(u + t(x-u)) \cdot t dt $$ where $f' = ...
2
votes
1answer
33 views

Iterated Integral and Sign Change in Answer

Given the iterated integral $\int_0^1\int_x^{2-x}(x^2-y) \, dy \, dx$, the value for the type I integral is, \begin{align*} & \int_0^1\int_x^{2-x}(x^2-y)\,dy\,dx \\ = {} & \int_0^1 ...
0
votes
0answers
43 views

Change of variable and partial derivative

Let us suppose we consider the following change of variables $(t,r)\rightarrow (T,R)$ with $$ f(t,r) \frac{\partial}{\partial t}= \frac{\partial}{\partial T} \quad(1)\\ g(t,r) \frac{\partial}{\partial ...
0
votes
1answer
26 views

Compute Surface Integral

Integrate $x^2+y^2$ over the upper hemisphere of radius $a>0$ centered at $(0,0,0)$. $\textbf{Edit}$ Consider the parametrization of the upper hemisphere given by $$X(\phi, \theta) = (a ...
0
votes
2answers
34 views

Show the function is continuous in $\Bbb R^2$

Show that the function f is continuous in $\Bbb R^2$ : $$f(x,y)= \left\{ \begin{array}{ 1cc} \sin(xy)/xy & xy\neq0\\ \\ 1, & xy =0 \\ \\ ...
2
votes
1answer
36 views

How to find this kind of function?

I am trying to find a function $f(x,y)$ with $f:\mathbb{R}^{2} \rightarrow [a, b]$ where $[a, b]$ is equal to $[-1, 1]$, $[0, 1]$, or some other small interval (open intervals are fine as well). The ...
-1
votes
2answers
41 views

Continuous derivative vs Continuous partial derivatives

Firstly, suppose $f:\mathbb R^n\to\mathbb R^m$ has all continuous partial derivative. I believe I have proved that this imply continuous derivative. Please tell me if this is actually true. For its ...
2
votes
0answers
33 views

How to find derivative of $\left\Vert x-a\right\Vert ^{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}$?

Here is what I've tried: $\alpha\left(x\right)=\sum_{i=1}^{n}\left(x_{i}-a_{i}\right)^{2},\ \ \beta\left(y\right)=\sqrt{y}, \ \ \ \gamma\left(z\right)=z^{p}$. Clearly: $\left\Vert x-a\right\Vert ...
1
vote
1answer
31 views

How to prove this identity in vector calculus (suffix notation)?

Let $\epsilon_{ijk}$ be the alternating tensor defined by $$\epsilon_{ijk} = \begin{cases} 0, & \text{if any of $i$, $j$, $k$ are equal}\\ 1, & \text{if $(i,j,k)=(1,2,3)$, $(2,3,1)$ or ...
0
votes
1answer
33 views

$\frac{\partial u}{\partial x} \cos \theta + \frac{\partial u}{\partial y} \sin \theta = \frac{\partial u}{\partial r} (z_0 + r e^{i\theta})$?

Let $u$ be a function of two variable and all its partial derivative exists and fix $z_0 \in \mathbb C $ and $r>0.$ My vague question: How to show: $\frac{\partial u}{\partial x} \cos \theta + ...
0
votes
1answer
13 views

Curvature at a point in a vector valued function

I am trying to determine the curvature when $t=2$ of the function $r(t)=<t^3,3t^2,8t>$ So I found $v(t)=<3t^2,6t,8>$ and $a(t)=<6t,6,0>$. So now that I have these two functions, I ...
1
vote
2answers
50 views

Minimizing a summation?

I have absolutely no idea how to approach this problem. I've been looking through notes, and I think I missed this when my professor discussed this in class. $$ \text{Consider the data}\\ i\: x_i\: ...
1
vote
4answers
118 views

Derivative of $\pmb{A}^T\pmb{x}$ with respect to $\pmb{A}$

Let $\pmb{A}$ be a real matrix and $\pmb{x}$ a vector such that $\pmb{A}^T\pmb{x}$ exists. Then how do I calculate the following result? $$\frac{\partial}{\partial \pmb{A} } \pmb{A}^T\pmb{x} = ?$$ Any ...
-1
votes
1answer
67 views

Continuity of a function on $\Bbb R^2$ [closed]

Function $f(x,y)$ is defined in a neighborhood of $(0,0)$. Then if for any t function $g(x) = f(x,tx)$ is continuous at $0$, then $f$ is continuous at $(0,0)$. if $f$ is continuous at ...
2
votes
3answers
39 views

Find maximum and minimum values of an equation on an elipse

I need some help with this. I've been struggling through this last chapter of my Calc III class, and I'm not sure how to do this (although, it doesn't seem like it should be difficult to do) $$ ...
-1
votes
1answer
30 views

Limit of a multidimensional function

Limits problem again... Is there an easy and fast way to calculate $$\lim_{(x,\,y)\to 0} \frac{\sin(x)}{y}$$ and $$\lim_{(x,\,y)\to 0} \frac{\sin(x)}{y} \cdot \frac{1}{||(x,y)||_2}?$$ Thanks!
2
votes
0answers
16 views

Existence of full-rank function with given components

Assume that $U \subseteq \mathbb{R}$ is open and $f_1,...,f_k : U \rightarrow \mathbb{R}^n$ are differentiable functions such that $(f_1,...,f_k)$ has rank $k$ everywhere. Can we find more functions ...
0
votes
1answer
34 views

Differentials of Multivariable Functions

A soft drink can is h centimeters tall and has a radius of r cm. The cost of material in the can is 0.0015 cents per cm$^2$ and the soda itself costs 0.002 cents per cm$^3$. The cans are currently 10 ...
3
votes
1answer
43 views

Inverse Function Theorem, Spivak's Proof

I'm having a lot of trouble following the proof of the following theorem. This is from Spivak's Calculus on Manifolds. 2-11 Theorem (Inverse Function Theorem). Suppose that $f: \mathbb{R}^n \to ...
2
votes
2answers
45 views

$\nabla \times \left(\frac{\mathbf{A \times r}}{r^3}\right)$, where $\mathbf{A}$ is independent of $\displaystyle\nabla \times$

The curl is just over $\mathbf{r}$ and $r$. I've been trying to pull the vector $\displaystyle \mathbf{A}$ out of the way, in order to get a expression much easier to deal with, but I have no idea how ...
1
vote
2answers
39 views

The derivative of a recurrence relation of functions

I am unsure of how to take the derivative of a recurrence relation of functions. For example consider the following recurrence relation: \begin{equation} \left\{ \begin{array}{cl} f_n(x) &= ...
3
votes
3answers
92 views

Points on Surface, Distance Optimized

How do I find the points on the surface: $$x^3+y^3+z^3=1$$ such that the distance to the origin is minimized? My Thoughts: Perhaps we can minimize the distance squared? Not sure.
2
votes
1answer
58 views

Is every compact set in $\mathbb R^2$ a continuous image of some compact set of $\mathbb R$?

Is it true that for every compact subset $A$ of $\mathbb R^2$ , there exist a compact set $B$ in $\mathbb R$ such that there is a continuous surjection from $B$ to $A$ ?
2
votes
0answers
41 views

Extensions of $C^k$ functions to the boundary [closed]

Assume $\Omega \subset \mathbb R{^n} $ is an open connected smooth domain. I have some propositions that I guess they are correct , but I want to be confident. If $f\in C_0^k(\Omega) $ then $f\in ...
2
votes
1answer
30 views

How to express curvature of a level set in terms of derivatives of a function?

Suppose I have a smooth function $u:\mathbb R^n\to\mathbb R$. Assume that its gradient doesn't vanish (near any point where we investigate it). Is there a list of different (intrinsic and extrinsic) ...
0
votes
2answers
17 views

Fidning multivariable limit

I'm having trouble in finding multi-variable limit and hope that an example like this could get me started on my work $$\lim \limits_{(x, y) \to (2,0)} \frac{1-cosy}{xy^2}$$
2
votes
2answers
33 views

$\displaystyle\iiint_E (x²+y²) \;\mathrm{d}V$ where $E$ is the region between the spheres $x^2+y^2+z^2 = 4$ and $x^2 + y^2 + z^2 = 9$

To be honest I'm not even too sure of what I'm integrating. I'm picturing two spheres touching each other, with a cylinder of two different radii going from the center of one to the other and I'm ...
0
votes
2answers
27 views

Derivative of dot product?

What's the derivative ${\partial \over \partial x} \langle x, f(x)\rangle$? According to the product rule it should be $1\cdot f(x) + x \cdot f'(x) $ but in my previous post I was told that this ...
1
vote
1answer
48 views

Any real valued continuous function from a closed bounded set in $\mathbb R^2$ is bounded and attains its bounds

Without using the idea of compact or sequential compactness , can we prove that if $A$ is a closed bounded set in $\mathbb R^2$ , then any continuous function $f:A \to \mathbb R$ is bounded and ...
0
votes
2answers
32 views

Critical points of $(x^2+y^2)e^{y^2-x^2}$

$\frac{df}{dx} = 2xe^{y^2-x^2}(1-x^2-y^2) = 0.$ $\frac{df}{dy} = 2ye^{y^2-x^2}(1+x^2+y^2) = 0.$ So, $2xe^{y^2-x^2}(1-x^2-y^2) = 2ye^{y^2-x^2}(1+x^2+y^2)$. $x(1-x^2-y^2) = y(1+x^2+y^2)$ $x-x^3-xy^2 ...
0
votes
1answer
34 views

Show that the function $f(x,y) = |xy|$ is differentiable at 0, but is not of class $C^1$ in any neighborhood of 0.

The problem from Munkres' *Analysis on Manifold is that Show that the function $f(x,y) = |xy|$ is differentiable at 0, but is not of class $C^1$ in any neighborhood of 0. My thought on the first ...
2
votes
3answers
66 views

Help finding specific book

I'm studying Engineering and I'm in my second year, studying Multivariable Calculus, but my University is kind of hard teaching me fresh calculus with topology and analysis, and is kind of hard, so I ...
1
vote
0answers
25 views

Vector Differential in specific form

for the operator: $\Delta A(t) = A(t+1)-A(t)$, let : $$ \Delta C = \sum^{T-1}_{t=0} [~~H(\pi(t+1)~|~\mu(t+1))~~ - ~~H(\pi(t)~|~\mu(t+1))~~] $$ Where $H (\pi(t+1)|\mu(t+1)) = \sum^n_{i=1}\pi_i(t) \log ...
0
votes
2answers
34 views

Taking derivative of function $g: \mathbb{R} \to \mathbb{R}$ defined in terms of $f: \mathbb{R}^{n+1} \to \mathbb{R}$.

Suppose we are given $g(r): \mathbb{R} \to \mathbb{R}$ where $g(r) = f(ry, r^2s)$ for $f: \mathbb{R}^{n+1} \to \mathbb{R}$ where $y \in \mathbb{R}^n, s \in \mathbb{R}$. How do we determine ...
1
vote
3answers
52 views

Critical points of $f(x,y) = \sin x \sin y, -\pi<x<\pi, -\pi<y<\pi$

$\frac{df}{dx} = \cos x\sin y = 0$ $\frac{df}{dy} = \cos y\sin x = 0$ $\cos x\sin y = \cos y\sin x$ $\frac{\cos x}{\sin x} = \frac{\cos y}{\sin y}$ $\cot x = \cot y$ $P_1 = (\pi/2,\pi/2)$ $P_2 = ...
0
votes
0answers
39 views

Vector Calculus - Polar Co-ords

I am having a lot of difficulty finding an approach to solving the following question: A dyon is a particle with both electric and magnetic charge; in suitable units $$\mathbf{E} = ...
0
votes
1answer
25 views

Multivariable Chain Rule for partially differentiable maps

The following statement is a simple consequence of the multivariable chain rule: Assume the subsets $X \subset \mathbb{R}^m$ and $Y \subset \mathbb{R}^n$ are open. Consider the maps $$g:X \to ...
1
vote
1answer
44 views

If $g(x,y)=(x+f(y),y+f(x))$ Why this function $g$ is onto?

Let $f:\mathbb{R}\to\mathbb{R}$ of class $C^1$ such that $|f'(x)|\leq b<1$ for all $x\in\mathbb{R}$. If we define $g:\mathbb{R}^2\to\mathbb{R}^2$ by $g(x,y)=(x+f(y),y+f(x))$ then why is $g$ an onto ...
4
votes
3answers
57 views

Is $f$ bijective on$\mathbb{R}^2$?

Let $f(x,y)=(x^2-y^2,2xy)$ a function from $\mathbb{R}^2\to\mathbb{R}^2$. Study if $f$ does have an inverse in whole $\mathbb{R}^2$? My approach: Since ...
2
votes
2answers
36 views

Limit of a function with 2 variables

I am given this function: $$f(x,y)=\begin{cases}\frac{xy^3}{x^2+y^4} & \text{ for } (x,y)\not=(0,0)\\ 0 & \text{ for } (x,y)=(0,0)\end{cases}$$ and I have to check if it is continuous in ...
1
vote
1answer
67 views

Correct order of taking dot product and derivatives in spherical coordinates

I tried to derive definition of divergence in spherical coordinates from gradient and got: $${\vec \nabla \cdot \vec A=\bigg (\frac{\partial}{\partial r}\hat r+\frac{1}{r}\frac{\partial}{\partial ...
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0answers
14 views

Manifolds Product

I was given the following exercise: Show that if $M$ is a $k$-manifold without boundary in $\mathbb{R}^m$, and if $N$ is an $l$-manifold in $\mathbb{R}^n$, then $M×N$ is a $k+l$ manifold in ...
3
votes
1answer
44 views

Bivariate infinite series: explicit sum?

Let $S_n(x,y)=\sum_{k=0}^n\binom{n}{k}\frac{x^k}{k!}y^{n-k}$, and consider the series $S(x,y)=\sum_{n=0}^\infty S_n(x,y)$, where $x,y\in \mathbb{R}$. My question is: does this series have an explicit ...
0
votes
0answers
30 views

Find all critical points of $f(x,y) = \frac{-x^3}3 + x - y^2$ and state maximum, minimum, or saddle points.

I'm confused on how to approach this question: Find the critical points and say whether they are maxima, minima or saddle points $$f(x,y) = -\frac{x^3}{3} + x - y^2$$ I have $$f_x = -x^2 + 1$$ ...
2
votes
1answer
38 views

Computing the shape operator

I am trying to compute the shape operator and Gaussian curvature for some smooth zero sets of polynomials $f$ in $\mathbb{R}^n$, oriented by $N = \nabla f / || \nabla f||$ The approach I thinking ...
2
votes
2answers
45 views

Minimize Total Cost of Box

So there is a rectangular box that has a volume of $8 m^3$. The top and bottom of the box is made with some material that has a cost of $8$ dollars per square meter. The sides are made with another ...
0
votes
2answers
32 views

Maximum/Minimum on a Unit Disk

I need to find absolute maximum and minimum values of $$f(x,y)=4x^3+3y^2$$ on a unit disk $D={(x,y)|x^2+y^2\le 1}$ I thought about finding $f_x$ and $f_y$ first to find in the critical point is in ...
0
votes
1answer
36 views

Finding Critical Points and Local Maxima/Minima or Saddle Point

I need help to find critical points of the function: $$f(x,y)=\frac{-x^3}{3}+x-y^2$$ Then I have to classify these critical points as local maxima/minima or saddle points. I thought that to find the ...
0
votes
0answers
24 views

Multi-Variable Calculus: Change variables in order to use a specific region for integration

After reviewing change of variables, I realized that every text I read provided the change of variable equations in the problem descriptions. From what I understand, the main use of changing variables ...
1
vote
3answers
57 views

Chain rule for integrals, how?

Can you please give some hints how to solve such a task: Given 3 smooth functions: $f: \mathbb R^2 \rightarrow \mathbb R$, $a,b: \mathbb R \rightarrow \mathbb R$. I should determin the derivative ...