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
14 views

continuous differentiability of a multivariable function

I have a function defined ad $Y=F(X_1,X_2,...,X_N)$, I want to prove that $F$ is continuously differentiable over $X$. Is there any theorems I can use? I tried to calculate the Jacobian matrix and ...
3
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1answer
46 views

Study of differentiablity of function

Study the differentiability of the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ $f(x,y)=\begin{cases} \frac{x^3+y^3}{x^2+\left|y\right|} & (x,y)\ne(0,0) \\ 0 &(x,y)=(0,0) \\ ...
1
vote
1answer
39 views

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$? Well, in the answer is no. it is written that $e^{x+y}$ for every $(x,y)$ has ...
1
vote
1answer
30 views

Complete a proof that $F(x,y)$ is contracting.

Can anyone fill in the dots in this proof? Let $D := [0,\frac{1}{2}]^2$. Show there is exactly one $(x,y)=(x^*,y^*)\in D$ such that \begin{align*} x &= \frac{x^3}{2} + y^4 + \frac{1}{4} \,, \...
1
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1answer
23 views

Using a gradient to calculate the minimum slope

given the function: $$z=f(x,y)=e^{-x^2-2y^2}$$ I'd like to find a point where if I were to place a ball, it would roll towards the direction $(2,1,a)$ . Also, at which point could I place the ball ...
1
vote
1answer
20 views

Local coordinates for Cylinders

Suppose point $A$ has intrinsic local coordinarcs of $(0,0)$ on a cylinder of radius $7$ and point $B$ has intrinsic local coordinarcs of $(6 \pi,4)$. Find two angles that spiral geodesics could form ...
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0answers
9 views

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=0.$$ ...
3
votes
2answers
46 views

Prove that a function is contractive

I'm stuck with the following. I need to prove that in $D:=[0,1]\times[0,1]$ the function $F$ is contractive, where $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is defined as: \begin{align} F(x,y):=(\frac{...
0
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0answers
9 views

Divergence of Material Derivative

Let $u : \Bbb{R}^n\times \Bbb{R} \to \Bbb{R}^n\times \Bbb{R} $ be a divergence free vector field. Then the material derivative $D $ is given by: $$ \frac { \partial u_j}{\partial t}+\sum_{i=1}^{n} ...
-2
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0answers
31 views

How to calculate derivative of two functions? [on hold]

If f(x,y) and g(x) are two functions and they intersect at a point (x*,y*) I want to evaluate that how this intersection point change with respect to x. Note that I cannot differentiate the x ...
0
votes
2answers
69 views

Is f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}} $with f(0,0)=0 continuous in (0,0) [duplicate]

I believe that the function: f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}}$ is continuous on the point (0,0) but i can't prove it. I know you have to choose something like $x=cy^{2}$(with c a constant) to prove ...
2
votes
2answers
43 views

Find $\lim_{(x,y)\to(0,0)} g \left(\frac{x^4 + y^4}{x^2 + y^2}\right)$ where $\lim_{z\to 0}\frac{g(z)}{z}=2.$

This limit seems different to me than all the other multi variable limits already asked on this site. Let $g \colon \mathbb R \to \mathbb R $ be such that $$ \lim_{z\to 0}\frac{g(z)}{z}=2. $$ ...
1
vote
1answer
28 views

Computing the Jacobian of the Euler equations

Given the Euler equations $$ \frac{\partial q}{\partial t}+\frac{\partial f(q)}{\partial x}=0,\qquad q=\begin{pmatrix}\rho\\\rho u\\\rho e\end{pmatrix}, \qquad f(q)=\begin{pmatrix} \rho u\\\rho u^2+...
3
votes
1answer
35 views

How is “expressing” a differential operator “in cylindrical coordinates” rigorously defined?

I'm a mathematician (with little knowledge of differential geometry) trying to study physics. One of the greatest problems is the language regarding coordinate transformations. I tend to think of such ...
1
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2answers
55 views

Directional Derivative help, solving for derivative = 0 when given constants

A function that is useful in studying the air flow over mountains is $$h(x,y) = \frac{h_0}{[(\frac{x}{a})^2+(\frac{y}{b})^2+1]^\frac{3}{2}} $$ where $h_0$, a, and b are all positive constants. (a) ...
1
vote
2answers
38 views

Using chain rule to find partial derivatives

Let $ r = \sqrt {x^2+y^2}$ and $\theta= tan^{-1}(y/x)$ be the usual polar/rectangular relationships. Furthermore, define $u(r(x,y),\theta(x,y)) = -sech^2(r)tanh(r)sin(\theta)$ and $v(r(x,y),\theta(x,...
2
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0answers
31 views
+150

Integral involving the von Mises-Fisher distribution

I'm going quickly through the VonMises-Fisher distribution $M$ on $\mathbb S^{d-1}$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $...
0
votes
2answers
27 views

How to use the implicit function theorem in this case?

Really hit a wall with this one: Prove that the equations: $$2x+y+2z+u-v-1=0\\xy+z-u+2v-1=0\\yz+xz+u^2-v=0$$ define around $(u,v,x,y,z)=(1,1,-1,1,1)$ a single function $\phi(u,v)=(x(u,v),y(u,v),z(u,...
1
vote
1answer
49 views

Calculate an integral with limit another integral

I have a list of integrals to do with a structure similar to this one, but I don't know how to attack anyone of them. I hope you can help me doing this one to understand how to do the other ones. ...
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0answers
13 views

Multivariable taylor series approximation

The function is of the form $$ F(X) = sum_{i=0}^n x_i*(c_i + ln(x_i/xt)) $$ where $ X = (x_1,x_2,x_3,...,x_n) $ $ xt = sum_{i=0}^n x_i $ $ c_i $ is a constant term for ith species I want to find ...
1
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2answers
40 views

Finding the shap of the volume $\int_0^{\pi/2}\int_0^{\pi/2}\int_0^{1} \left(\rho^2 \sin \phi \right) d \rho d \phi d \theta$

I need to find the shap of the volume:$$\int_0^{\pi/2}\int_0^{\pi/2}\int_0^{1} \left(\rho^2 \sin \phi \right) \,\mathrm d \rho \,\mathrm d \phi\,\mathrm d \theta$$ I thought that the shape is ...
1
vote
1answer
14 views

Gradient of function, which has codomain R^2 or bigger.

For example I have a function: $$f(x_1,x_2) = \begin{bmatrix} x_1x_2^2 + x_1^3x_2\\x_1^2x_2 + x_1 + x_2^3\\\end{bmatrix}$$ Is it possible to find a gradient of this function? Because knowing the ...
1
vote
1answer
18 views

What's the formula to map between multiindices and indices?

What is the formula to map between multiindices and indices? By multiindex, I mean a variable $I\in\mathbb{N}^d$ where $|I|=\sum\limits_{i=1}^d I_i=n$. Here, $d$ denotes the dimension. Basically, ...
0
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0answers
16 views

Question about calculus of variation.

What is the difference between finding maxima or mimima i.e. critical point of a function and calculus of variation?
4
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2answers
71 views

the uniform convergence of the sequence of functions

Let $f_1:[a,b]\rightarrow \mathbb{R}$ be a Riemann integrable function. Define the sequence of functions $f_n:[a,b] \rightarrow \mathbb{R}$ by $f_{n+1}(x)=\int_a^x f_n(t)dt,$ for each $n\ge 1$ and ...
1
vote
1answer
125 views
+100

Theorem regarding Change of Variables in finite dimesnion

My question is based on Change of Variables in Multiple Integrals II Peter D. Lax > It is not necessary to read the paper before answering this question.The author tried to prove change of variables ...
0
votes
1answer
37 views

Area of a domain with Stokes' Theorem

This question came up on a preliminary exam: Define $$g(s,t)=(x(s,t),y(s,t))=(\cos(s)+\cos(t),\sin(s)+\sin(t)),$$ on the region $-\pi<s<\pi$, $s<t<s+\pi$. (The function $g$ is one-...
-2
votes
1answer
69 views

Integral Optimization Problem [on hold]

A butterfly is flying in a room and the temperature of this room is given by the function $$T(x,y,z) = 3x^2+y^4+2z^2$$ (in Celsius °C). The butterfly is at the point $(1,1,1)$ and she realizes that ...
0
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0answers
27 views

$N$-dimensional volume (of revolution)

Consider the system of coordinates $\{x_{1},x_{2},...,x_{n}\}$ and an n-dimensional shape such that, in $\{x_{1},x_{n}\}$ (and $x_{2}=x_{3}=...=x_{n-1}=0$) it is inside the lines $x_{n}=ax_{1}+b$ and $...
0
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1answer
60 views

curve between two points having maximum area [on hold]

Find the curve(i.e. the segment of a standard curve like circle, ellipse etc.) amongst all curves(segments) that have fixed total length, passes through $ (a,b)$ and $(c,d)$ and has maximum area ...
1
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2answers
48 views

Question about proof: continuity of partial derivatives implies total differentiability

I have a lack of understanding regarding this proof, and since the proof is not in English, I will simply write it down up to the point where I can't go further: Statement: Assume $U \subset \Bbb R^...
2
votes
1answer
55 views

Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: \begin{equation} f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
0
votes
3answers
85 views

Evaluate $\iiint dx\,dy\,dz$ betweem $x=0,y=0,z=0, x+y+2z=2$

I need to evaluate $$\iiint dx\,dy\,dz$$ the volume between $x=0,y=0,z=0, x+y+2z=2$ I am stuck about choosing the limits of integration, I think that the limits should be: $$\int_0^{(2-x-y)/2}\...
3
votes
3answers
136 views

Prove $\lim_{(x,y)\to (0,0)} \frac{x^2 y^3}{x^4 + y^4} =0$ without $\varepsilon - \delta$.

Unlike Multivariable Delta Epsilon Proof $\lim_{(x,y)\to(0,0)}\frac{x^3y^2}{x^4+y^4}$ --- looking for a hint I would like to avoid the $\varepsilon - \delta$ criterium. Prove $$\lim_{(x,y)\to (0,0)...
2
votes
1answer
86 views

Second Differential

Let $(x,y,z)$ a coordinate system, $M=\mathbb{R}^3$ and we also denote by $x$ the first coordinate function : $x:M \rightarrow \mathbb{R},\; q=(a,b,c) \mapsto a$. We have $dx:TM \rightarrow \mathbb{R}...
-2
votes
1answer
21 views

Distance between a point and a conic curve

I have a point $r=(100,0)$ and want to find the closest point to it from this set: $$k = \{(a,b) : b^2=1+a/4\}$$ where $a$ belongs to $[-4,0]$. I thought about defining function $h(x)=|r-x|$, and ...
4
votes
1answer
47 views

Estimate the value of f at a given point

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a differentiable everywhere. Assume $f(-\sqrt2,-\sqrt2)=0$, and also that $|\dfrac{\partial f}{\partial x}(x,y)|\le |\sin(x^2+y^2)|$ and $|\dfrac{\...
0
votes
1answer
38 views

Finding Partial Derivative in two ways

I am supposed to find $f_x(0,0)$ of $\frac{5x^2y}{x^4+y^2}$, EDIT: which has a defined value of $0$ at $(0,0)$. The way I did it, I first found the general expression for $f_x(x,y)$, which is $$f_x(...
0
votes
1answer
42 views

Flux of a vector field $F(x,y,z)$

I have a circle in the $yz$-plane centered at $(0,2,0)$ with radius $1$. The surface $\Sigma$ is obtained by rotating the circle around the $z$-axis. I want calculate the flux of the vector field $...
3
votes
2answers
47 views

Are there more types of critical points beyond maxima/minima/saddle points for higher dimensions?

I had a course on single variable calculus and at that point, we had minima and maxima. Now on several variables calculus, there is maxima, minima and saddle points. Certain books of several variables ...
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0answers
22 views

Help needed in deriving the complex root of raised cosine power spectrum

The raised cosine power spectrum (energy spectrum) is defined as $$|\hat{\phi}|^2 = \begin{cases} 1,& \text {if} |\omega| \leq \pi (1-b)\\ \dfrac{1}{2}[1+\cos\dfrac{|\omega|-\pi(1-b)}{2b}],& \...
0
votes
4answers
65 views

Using Double Integral Find the volume of sphere $x^2 + y^2 + z^2= 4 $ cut by cylinder $\ x^2+y^2=2y $

Using Double Integral Find the volume of sphere $x^2 + y^2 + z^2= 4 $ cut by cylinder $\ x^2+y^2=2y $ , When i try to make integral the limits are: $\ -1<= x<=1 $ and $\ 0<=y<=2 $ ,but i ...
0
votes
1answer
33 views

Normal vector on a plot

Do a sketch of $f$ with the equation $f(x,y)=0$. Give in all non singular points of the curve a normal vector. $f(x,y)=x^{3}-x-y$ How can I do this thing with normal vector? I know that singular ...
1
vote
1answer
20 views

The gradient of a convex function is controlled by its oscillation on a larger ball

My problem is Let $f:\mathbb R^n\longrightarrow R$ be a convex function. Knowing that $$|\nabla f(x)|=\sup_{y\neq x}\frac{[f(x)-f(y)]^+}{|x-y|}$$ ($[f(x)-f(y)]^+$ represents $\max\{[f(x)-f(y)],...
0
votes
0answers
53 views

Stokes theorem for the flux through a surface

I have a surface $\phi(u,v)=(u\cos v,1-u,u\sin v) ; u\in[0,1]; v\in[0,2\pi]$. I want calculate the flux through $\phi$ of $F(x,y,z)=(z+\arctan y,\frac{x^5}{1+z^2},x^2ze^{y^2}) $ as $\int_{\phi} <...
0
votes
3answers
42 views

Is $|xy|$ differentiable in $(0,0)$?

Is $f(x,y) = |xy|$ differentiable in $(0,0)$? I have no idea how to approach this problem.
1
vote
1answer
57 views

Evaluate $\int_{\theta=0}^{2\pi}\int_{r=0}^{1/\sqrt 2}\int_{z=r}^{\sqrt{1-r^2}}r\text d z \text d r\text d \theta$

I need to evaluate $$\int_{\theta=0}^{2\pi}\int_{r=0}^{1/\sqrt 2}\int_{z=r}^{\sqrt{1-r^2}}r\text d z \text d r\text d \theta$$ My attempt: $$\int_{\theta=0}^{2\pi}\int_{r=0}^{1/\sqrt 2} \left(...
0
votes
3answers
58 views

How to draw the plane $x+y+2z=2$

I need to evaluate $$\iiint \text d x \text d y\text d z$$ the planes are $x=0,y=0,z=0,x+y+2z=2$ Is there a method to draw the plane? it is easy to draw $x=0,y=0$ and $z=0$ but how can I draw to ...
0
votes
0answers
22 views

Stationary points [duplicate]

Why is the following statement false? "A stationary point of a function of 2 variable is ALWAYS either un local maximum, local minimum or a saddle point"