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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3
votes
1answer
155 views

If $f$ is twice differentiable, $(f(y) - f(x))/(y-x)$ is is differentiable

Suppose $f: \mathbb{R} \to \mathbb{R}$ is a $C^{1}$ function. Then, define a new function $F: \mathbb{R}^{2} \to \mathbb{R}$ by: $$ F(x,y) = \begin{cases} \displaystyle \frac{f(y) - f(x)}{y - x} ...
2
votes
1answer
68 views

Finding a volume

Find the volume of $D\{(x,y,z)\in \mathbb{R}^3:\frac{x^2}{a^2} +\frac{y^2}{b^2}\leq z\leq 1 \}$ It looks like (1) I believe this could be solve with a double integral an considering the ...
1
vote
1answer
29 views

Potential in $2$ dimensional systems

Given a $1$ dimensional dynamical system represented by $\dot{x}=f(x)$ we define the potential $V(x)$ to be the function that satisfies $\dot{x}=f(x)= -\frac{\partial V(x)}{\partial x}$. How to we ...
0
votes
3answers
2k views

Finding volumes - when to use double integrals and triple integrals?

This is not a technical question at all, but I'm quite confused about what should I use to compute volumes in $\mathbb{R}^3$ with integration. I've read somewhere that a double integral gets the ...
-2
votes
1answer
56 views

Need a translation: Mathspeak to english

Watching a video on multiple integration. Maybe its that the coffee has not kicked in but I am having trouble with understanding the graphing term "mapping". Can anyone put it in layman's terms or at ...
0
votes
1answer
36 views

Question about limit.

My question is that in my practical sheet I have been given a question which says show that limit doesn't exist and question is $f(x,y)= \frac{x^2}{x^2+y^2-x}$ s.t $(x,y)\ne(0,0)$ My question is: ...
0
votes
1answer
57 views

Differentiation with help of Frenet Frame

Show that if $(\frac{1}{k})^{'} \neq 0$ and $(\frac{1}{k})^2 + ((\frac{1}{k})^{'}\frac{1}{\tau})^2$ is a constant, then a unit speed curve $\alpha$ lies on a sphere. Using the following formulas ...
3
votes
1answer
82 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
1
vote
1answer
22 views

differential split of $2$ variable problem

$d(x_1x_2)= x_1dx_2 + x_2dx_1$ as given in theory now $\int d(x_1x_2)= \int x_1dx_2 + x_2dx_1$ but integrating both the sides give $x_1x_2 = 2x_1x_2 ..$ why ? I guess I am missing something very ...
0
votes
1answer
27 views

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ [duplicate]

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ $\displaystyle \int_1^2\int_0^1\frac ...
2
votes
3answers
84 views

Why is the value of this line integral constant

Consider the line integral given by $$\int_C \frac{(x+y)\,dx-(x-y)\,dy}{x^2+y^2}$$ where $C$ is any simple closed curve around the origin. Can someone explain, without using complex analysis, why this ...
1
vote
1answer
74 views

How is statistical uncertainty calculated for the modulus function?

I know it's an unusual function to calculate an uncertainty for, but I haven't been able to figure out a reasonable means for calculating derivatives for it to do so myself. I know modular arithmetic ...
1
vote
1answer
27 views

Proof for $\int_{S^n(r)} \, dx=\int_{S^n(1)}r^n \, dx$?

I read the wikipedia article about $n$-sphere. I'm trying to give a proof for the following formula $$ \int_{S^n(r)}dx=\int_{S^n(1)}r^ndx,\tag{*} $$ where $S^n(r):=\{x\in {\Bbb R}^{n+1}:|x|=r\}$ for ...
2
votes
2answers
628 views

Multidimensional Fourier transform of the laplacian

In my course on electromagnetic field theory we use the Fourier transform to simplify Maxwell's equations, for example: $$\frac{\partial ^2\vec E(\vec r,t)}{\partial t^2} \rightleftharpoons ...
3
votes
2answers
146 views

Double integral — tricky?

If $f(x,y) = x^2+y^2$ and $D=\{(x,y)\in\mathbb{R}^2:x^2+y^2\geq1, x^2+y^2-2x\leq0 \text{ and } y\geq0\}$, find $\displaystyle\int\displaystyle\int_D f$. $D$ looks like the intersection between ...
0
votes
1answer
43 views

Simple integration of a differential along 3 separate paths

So I have $$dw = \frac{y}{a}dx + \frac{x}{a}dy$$ and points $$A=(0,0), B=(1,0), C=(1,1), D=(0,1)$$ How do I integrate along paths ABC and ADC? and how can I change variables integrate along the ...
3
votes
1answer
102 views

Find the critical points for $F(x,y,z)=-x^{3}-y^{2}+2xy+x+2z$

I started by taking the first order partial derivatives: $F_{x}=-3x^{2}+2y + 1$ $F_{y}=-2y+2x $ $F_{z}=2 $ Now I would try to solve it for $F_{x}=F_{y}=F_{z}=0$ but $F_{z}=2$. How can I proceed or ...
0
votes
1answer
78 views

Mapping Confusion -Implicit Function Theorem-

Here is the Implicit Function Theorem statement: "Let $g : R^k \times R^n \to R^n$ be a continously differentiable function s.t. $g(x_0, y_0) = c$ and $D_yg(x_0,y_0) : R^n \to R^n$ is an isomorphism. ...
3
votes
2answers
184 views

Solution of a partial differential equation.

Find $u$ if $$\dfrac{\partial^2 u}{\partial x^2} = 6xy, \,\,u(0,y) = y, \,\,\dfrac{\partial u}{\partial x}(1,y)=0.$$ I have tried by laplace transformation $$\displaystyle ...
0
votes
3answers
181 views

Derivatives of multivariable functions

I would like to make a few statements about a simple object - the derivative of a univariate function - and apply and relate its features and my understaning of them to multivariate functions. ...
1
vote
1answer
69 views

Arclength does not change with reparametrization

Recall that the length of a curve $\alpha : [a,b] \rightarrow \mathbb{R}^3$ is given by $L(\alpha) = \int |\alpha'(t)| dt$. Let $\beta(r): [c,d] \rightarrow \mathbb{R}^3$ be a reparametrization of ...
2
votes
0answers
109 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
0
votes
3answers
54 views

A question about a simple integral.

How could I show that the $$\iint\sin(x)dxdy$$ along the domain $$x^2+y^2\leq1$$ is zero? I tried using polar coordinates but to no avail. Had thought about claiming Sine is an odd function so the ...
0
votes
2answers
253 views

degree of homogeneity

I have the function $$f(x,y)=\frac{y^b}{x^a}+\frac{x^b}{y^a}\quad a,b\gt0$$ The questions I have to answer are For which a and b is the function homogenous? Determine the degree of homogeneity My ...
2
votes
0answers
32 views

Proving a diffeomorphism that involves trigonometric functions.

If $f:\mathbb{R}^2\to\mathbb{R}^2, f(x,y)=(x\cos (y), \sin (x-y))$. Then $\exists X,Y\subset\mathbb{R}^2$ both open such that $(\pi/2,\pi/2)\in X$ and $f:X\to Y$ is a diffeomorphism. I'm not sure ...
0
votes
1answer
362 views

Find the volume of the solid bounded by $z=x^2, y=3, z=4$

Find the volume of the solid bounded by $z=x^2, y=3, z=4$ $4=x^2 \rightarrow x=\pm2$ $\displaystyle V=\int_{-2}^2\int_0^3x^2dydx=\int_{-2}^2yx^2\Big |_0^3dx=2\int_0^23x^2dx=2\Big[x^3\Big|_0^2=16$ ...
1
vote
1answer
48 views

maxima and minima of 2 variable function

How can I show that $f(x,y)=e^x cos(y)$ doesn't have maxima nor minima in the unit circle? Because $f_x = f_{xx} =0$ when $x=0$ and $y=\frac{\pi}{2} +n\pi, n\in \Bbb{Z}$. and isn't ...
0
votes
1answer
51 views

Compute $\int_0^2\int_{\frac y2}^1e^{x^2}dxdy$

Compute $\displaystyle \int_0^2\int_{\frac y2}^1e^{x^2}dxdy$ $\displaystyle\int_0^2\frac {e^{x^2}}{2x}{\Big|}_{\frac y2}^1dy=\int_0^2\frac e2-\frac {e^\frac {y^2}{4}}{y}dy$. I am having trouble ...
2
votes
2answers
183 views

Is Gradient really the direction of steepest ascent?

I want to intuitively understand why the gradient gives you the direction of the steepest ascent of a function. Apart from the already posted questions, my confusion arises from the fact that we form ...
2
votes
1answer
198 views

Derivative of $(Ax - b)^T(Ax-b)$

I am trying to take the derivative of $(Ax - b)^T(Ax-b)$ and setting it to zero without expanding the multiplication, by only using matrix calculus. I knew the partial derivative of $x^Tx$ according ...
0
votes
1answer
99 views

Advanced calculus

Let $Q(x)=\sum_{i,j=1}^{n} c_{ij}x_ix_j >0$ for every $x\neq 0$ where $c_{ij}=c_{ji}$ for $i,j=1,2,\ldots,n.$ Show that $$\int ...
0
votes
1answer
68 views

When to use the jacobian $\mathcal{J}$

I am having trouble with how and when to set up a Jacobian. For instance, if I need to find the integral of some function ${H\left(f(x),g(x)\right)dx}$. If I let $f(x)=u$ and $g(x)=v$ then I should, I ...
1
vote
1answer
28 views

“Doubling the variable” and differentiability

Let's consider a domain $E \in \mathbb{R}^d$ and a function $f(x,y): E \times E \to \mathbb{R}$. Suppose $f \in C^2(E \times E)$. If we define a function $g$ on $E$ by $g(x):=f(x,x)$, is it true that ...
2
votes
0answers
66 views

Proving that $f:U\to\mathbb{R}^n$ differentiable is an open map when $\det \operatorname{J}f(a) \neq 0$

Let $U\subset\mathbb{R}^n$ be an open set and $f:U\to\mathbb{R}^n$ a differentiable function such that $\det \operatorname{J}f(a)\neq 0\; \forall a\in U$. Prove that if $V$ is an open subset of $U$ ...
0
votes
1answer
52 views

Spherical Coordinates Representation

I just wanted to know what the set of all points in which spherical coordinates can be shown in more than one way is? I think it is only the origin but I am not sure
4
votes
0answers
38 views

Cone is a submanifold iff it is a vector subspace

Could you tell me how to prove the following? Let $\emptyset \neq M \subset \mathbb{R}^n $ be a cone ($\forall x \in M, t \in \mathbb{R} : tx \in M $ ), $0 \le d \le n.$ Prove that $M$ is a $d$ ...
0
votes
2answers
142 views

Proving or disproving that $\{(x,y) : xy > 0\}$ is open

Here's what I have so far: Let $D = \{(x, y):xy>0\} \subset \mathbb{R}^2$ and let $(a,b) \in D$ such that, WLOG, $a\leq b$. Let $\delta = |a|$. $$ (a,b)\in D \Rightarrow ab > 0 \Rightarrow (a ...
2
votes
1answer
32 views

Change linear plot to 100% plot in Wolfram Alpha

Recently I have used this input for WolframAlpha: Plot (forumla1), (formula2), (formula3), {a, 0, 50} It's generating "Linear Plot" like on picture on left. Is ...
3
votes
4answers
149 views

Evaluation of a particular type of integral involving logs and trigonometric function

Is there any closed form for $$ \int _0 ^{\infty}\int _0 ^{\infty}\int _0 ^{\infty} \log(x)\log(y)\log(z)\cos(x^2+y^2+z^2)dzdydx$$ if yes then how to prove it?
0
votes
1answer
40 views

Question regarding Continuity of F(x,y)

Let $f(x,y) = \begin{cases} \frac{2(x^3+y^3)}{x^2+2y}&\text{ } (x,y)\not=(0,0)\\ 0 &\text{ }(x,y) =(0,0). \end{cases}$ show that first order partial derivatives of $f$ wrt x and y exist at ...
0
votes
0answers
94 views

What is a transform?

I've been working in vain to find a way to find the integral of an intractable function. It's great practice anyway. I thought about using intergration by parts with three functions to solve it and ...
1
vote
1answer
32 views

Help with vectorial analysis exercise

Let $D(0,r) := \left\{ {x \in \mathbb{R}^n: \|x\| \leq r }\right\}$ and $f:D(0,r) \rightarrow \mathbb{R}$ be a continuous differentiable function in the interior of $D(0,r)$. I'm trying to show ...
2
votes
1answer
44 views

Find the volume of the solid bounded by $z=x^2+y^2+1$ and $z=2-x^2-y^2$.

Question: Find the volume of the solid bounded by $z=x^2+y^2+1$ and $z=2-x^2-y^2$. Setting the 2 equations equal w.r.t. $z$, $x^2+y^2+1=2-x^2-y^2 \rightarrow x=\pm\sqrt{\frac 12-y^2}$ Therefore the ...
1
vote
1answer
49 views

Compute $\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ $\displaystyle\int_0^1\int_1^2\frac{y}{x+y^2}dydx=\int_0^1\int_1^2y(x+y^2)^{-1}dydx$ How do I integrate the ...
2
votes
0answers
45 views

Continuity of the orthogonal matrix-valued function

Given $d<k$. Suppose that $H:\mathbb{R}^k\rightarrow {\cal M}(\mathbb{R})_{d,k}$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$. Now define a ...
1
vote
0answers
34 views

Separation of integrand for multivariate integration when integrand is a product of single variable functions

If $f(x)$, $g(y)$, and $h(z)$ are real-valued functions of a single variable, does the following always hold? Is this the case for numerical approximations of the integral using quadrature? $$ ...
1
vote
1answer
53 views

How should I prove $\operatorname{vol}_{n+1}B_{n+1}=\int_0^1 \operatorname{vol}_n S^n(r)dr$ without using spherical coordinates?

Let $B_n:=\{x\in{\Bbb R}^n:|x|\leq 1\}$ and $S^n(r):=\{x\in{\Bbb R}^{n+1}:|x|=r\}$. Then we have the following formula $$ \operatorname{vol}_{n+1}B_{n+1}=\int_0^1 \operatorname{vol}_n S^n(r)dr. ...
4
votes
2answers
83 views

Show that the given function is a diffeomorphism

Let $U=\{x\in\mathbb{R}^n: ||x||<1\}$. If we define $f:U\rightarrow\mathbb{R}^n$ by $f(x) = \displaystyle\frac{x}{\sqrt{1-||x||^2}}$, show that $f$ is a diffeomorphism and ...
0
votes
2answers
461 views

Finding a line L perpendicular to line T that passes through point P in R3?

The question asks: Find the line through $(3,1,-2)$ that intersects and is perpendicular to $$x = -1 + t, y = -2 + t, z = -1 + t.$$ My thoughts: Say the point of intersection is $(x_0,y_0,z_0)$, ...
2
votes
3answers
207 views

a question about double integral

Let $a,b$ be positive real numbers, and let $R$ be the region in $\Bbb R^2$ bounded by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Calculate the integral $$ ...