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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2
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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
150 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
95 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 $$ ...
3
votes
2answers
50 views

Solve Helmholtz equation

$$U_{xx}+U_{yy}+k^2U=0$$ Solve by separation of variables by assuming $u(x,y)=X(x)Y(y)$ with the following conditions: $$ U(0,y)=0,\,\, U(2,y)=0,\,\, U(x,0)=0,\,\, U(x,1)=0, $$ This is ...
1
vote
1answer
99 views

Proving a map between the triangle and the square is bijective

Prove that $$(u,v) \mapsto \left(\frac{\sin u}{\cos v}, \frac{\sin v}{\cos u}\right)=\left(x, y\right)$$ is a bijection between the interior of the triangle $T:= \{0\le u,v; u+v \le \frac {\pi}2\}$ ...
3
votes
0answers
99 views

What is the domain and range of this multivariable function?

What is the domain and range of the following multivariable function? $g(x,y,z) = {1\over \sqrt{(4 - x^2 - y^2 - z^2)}}$ g is real valued. So far I have that the domain is the set of a vectors ...
2
votes
2answers
270 views

Finding extreme values when the determinant of the Hessian at a critical point is zero.

We want to determine extreme values of $f(x,y)=x^3+xy^2-x^2y-y^3$. We first determine critical points by solving $\dfrac{\partial f(x,y)}{\partial x}=0$ and $\dfrac{\partial f(x,y)}{\partial y}=0$ ...
1
vote
1answer
76 views

Why is this true: $\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec v)+\vec v \cdot (\nabla \cdot S)$?

I am doing fluid mechanics and I don't understand a particular step that is being used. It is the following step which I don't understand: $\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec ...
7
votes
2answers
154 views

Mathematically, why does $[ma]\mathrm{d}x = [mv]\mathrm{d}v$?

I am taking an introductory level class, Physics with Calculus, using Priscilla Laws' Workshop Physics. The activity guide has asked me to prove that: $$ma\,\mathrm{d}x = mv\,\mathrm{d}v$$ My ...
1
vote
0answers
40 views

Integral of a determinant of Jacobian depends on the boundary values only

Let $B$ be the closed unit ball in $\mathrm{R}^n$ with the 2-norm. Let $\phi : B \to \mathrm{R}^n$ be smooth such that $\det D \phi = 1$ on $\partial B$. Why is $\int_B \det D \phi = \int_B 1$? In ...
1
vote
4answers
2k views

How do I find the orthogonal basis for this plane?

Question: P is a plane through the origin given by x + y + 2z = 0. Find an orthogonal basis v1, v2 ∈ P. My answer: I'm assuming the question asks for two vectors that span this plane P. But ...
1
vote
1answer
115 views

Multi Variable Calculus Question

Please help me with this multi-variable calculus question given a Taylor Polynomial of degree 1.
0
votes
1answer
49 views

What is the formula for expressing $f(x,y,z)$ in terms of second-order partial derivatives?

By the fundamental theorem of calculus $f(x) = f(0) +f'(0)x + \int_0^x \int_0^v f''(u)dudv$. What is the explicit formula (using integration) for expressing $f(x,y,z)$ in terms of $\frac{\partial^2 ...
5
votes
1answer
71 views

Find $\alpha$ such that the given point is critical for a implicitly defined funtion.

Can anyone check my solution for this exercise? Let $F:\mathbb{R}^3\rightarrow\mathbb{R}$ be given by $F(x,y,z) = \alpha xz + x\arctan(z) + z\sin(2x+y) -1.$ Prove that a function $z=f(x,y)$ ...
1
vote
1answer
236 views

Change of variable (translation) in complex integral

If I have a real integral, e.g. $\int f(x+2) \ dx$, I can substitute $y = x+2$, so $dy = dx$. But if my function is complex, am I still allowed to do this? In which cases I cannot apply a ...
0
votes
2answers
82 views

Does a plane have to be spanned by two vectors that are perpendicular?

I'm beginning to learn some vector calculus, and I am slightly confused about the textbook's explanation of planes spanned by two vectors. They said for example that the xy plane is an example of the ...
1
vote
1answer
32 views

Conditions for global invertibility of a function

Let $U$ be open and convex, $f:U \rightarrow \Bbb R^n$ continuously differentiable, where $\|Df(x)-Id\| < 1$, for all $x \in U$. Then, $f$ is injective on $U$ (thus, $f:U \rightarrow f(U)$ is ...
3
votes
0answers
110 views

Multivariable calculus along with tensors …etc to start studying General Relativity

I bought Spivak Calculus on Manifolds last time and I was really really disappointed... I opened the first chapters and I understood nothing of what he was saying. But i need to understand ...
2
votes
0answers
120 views

A nabla operator identity in ortho-normal coordinate systems

Let $u_1,~u_2,~u_3$ be components of the position vector $\vec{u}$: $$\vec{u}=u_1 \vec{e}_{u_1}+u_2 \vec{e}_{u_2}+u_3 \vec{e}_{u_3}$$ in an ortho-normal coordinate system (not necessarily Cartesian) ...
0
votes
3answers
24 views

Finding the vector norm

How would I solve the following problem. Find the value of $a$ such that $z=2i-j+ak$ is three times as long as $b=2i+j-k$ would this mean that the value of a=3 so that k is three times as long in a ...
1
vote
1answer
38 views

Calculating a tangent line to an implicitly given function

Let $F:\mathbb{R}^3 \to \mathbb{R}^2 $ be continuously differentiable and let $p\in\mathbb{R}^3 $ for which $F(p)=0$ . Assume $rank DF|_p =2 $ and denote by $E$ the set $E=(x\in \mathbb{R}^3 | ...
2
votes
1answer
72 views

Implicit Function Theorem [Understanding theorem in book]

"Let $\mathbb{F}$ be a real-valued continuously differentiable function defeined in a neighborhood of $(X_0, Y_0) \in \mathbb{R}^2$. Suppose that $\mathbb{F}$ satisfies the two conditions: ...
0
votes
1answer
32 views

Implicit Function Theorem [Finding $\frac{du}{dx}$]

In the link: http://www2.imperial.ac.uk/~svanstri/Files/multivariable-calculus.pdf on the very last slide, I am confused on on why they are doing the jacobian matrix $\begin{pmatrix}\frac{du}{dx} \\ ...
0
votes
1answer
123 views

A holomorphic function is conformal

I am trying to show that if a function $f = u+iv$ is holomorphic with $\partial_z f(z)$ always non zero, then $f$ is a conformal mapping, i.e. it preserves angles between smooth curves. If $f$ is ...
0
votes
1answer
71 views

Curvature of plane curve

This is a motivation for the definition of the curvature of a plane curve: Suppose then that $\gamma$ is a unit-speed curve in $\mathbb{R}^2$. As the parameter $t$ of $\gamma$ changes to $t+\Delta ...
0
votes
2answers
58 views

Conservative fields and not simply connected domains

I'm given a continuously differentiable $f:\mathbb{R}\to\mathbb{R}$ , we define a function $\vec{F}:\mathbb{R}^2 \to \mathbb{R}^2 $ by: $\vec{F}( \vec{x} ) =f(|| \vec{x} || ) \vec{x} $ . The question ...
1
vote
1answer
76 views

Area of the portion between a solid cylinder and Surface $z=x^2-y^2$

Find the area of the portion of the surface $z=x^2-y^2$ in $\mathbb{R}^3$ which lies inside the solid cylinder $x^2+y^2\le1$. I parametrized the surface as ...
0
votes
1answer
25 views

Maximum value of directional derivative is invariant

Let $\vec{a}, \vec{b} , \vec{c} $ be three vectors in $\mathbb{R}^3 $ that are orthogonal to each other and have length 1. Show that for every differentiable $f:\mathbb{R}^3 \to \mathbb{R}$ and for ...
1
vote
1answer
39 views

Extrema on a given set

Could you tell me if my approach to finding extrema on a set is good? Let's take a function $f(x,y,z)=x+y+z$ and a set $N= \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 \le z \le 1 \} = \{(x, y, z) \in ...
2
votes
1answer
168 views

Solving exercise with Leibniz rule

I'm asked to prove that if $f(x) = \left(\displaystyle\int_0^x e^{-t^2}dt \right)^2$ and $g(x) = \displaystyle\int_0^1 \displaystyle\frac{e^{-x^2(t^2+1)}}{t^2+1}dt$ then $f'(x)+g'(x)=0$ and conclude ...
1
vote
1answer
122 views

Fubini Theorem. How it can be?

How this equality is true: $\left ( \int\limits_{-a}^a e^{-x^2} dx \right )\cdot \left ( \int\limits_{-a}^a e^{-y^2} dy \right )=\int\limits_{-a}^a \int\limits_{-a}^a e^{-(x^2+y^2)}\,dx\,dy$ For ...
-1
votes
2answers
1k views

How do you find the acute angle between the lines: $x+2y=7$ and $5x-y=2$. [closed]

Find the acute angle between the lines: $x+2y=7$ and $5x-y=2$. Use vectors.
1
vote
3answers
59 views

Determine $\lim_{(x, y)\to (0, 0)}\frac{-x+y+1}{x^2-y^2}$

In wolframalpha I tried to calculate $$\lim_{(x, y)\to (0, 0)}\dfrac{-x+y+1}{x^2-y^2}$$ and it returns : (limit does not exist, is path dependent, or cannot be determined) can't we say directly that ...
1
vote
2answers
147 views

Limit of $f(x,y) = (\sin(x) - \sin(y) )/ (\tan(x) - \tan(y))$

I'm asked to check if there exists $\lim_{(x,y) \to (0,0)} f(x,y)$ for the following function: $$ f(x,y) = \begin{cases} \displaystyle \frac{\sin(x) - \sin(y)}{\tan(x) - \tan(y)} & \text{ if } ...
2
votes
0answers
191 views

Find maximum of a double integral over a region

I have a region given by $$R = |{ax}|+|{by}| \le 1$$ and $$f(x,y) = \iint\limits_{R}{(ax-by)^2*(3ab^3+12a^3b-6a^3b^2)*\sin^2({\pi ax + \pi by}})dxdy$$ I need to find the values of $a$ and $b$ that ...
0
votes
1answer
42 views

Chain rule for the $\frac{d}{d\mathbf{w}}\exp(-|\mathbf{w}|^2)$: What about the absolute value?

My question is about the vector derivative $\frac{d}{d\mathbf{v}}\exp(-|w|^2)$. If I apply the chain rule I get $-2\mathbf{w}\exp(-|\mathbf{w}|^2)$ , is this correct or can I leave out the absolute ...