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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1answer
82 views

Boundary integral of a harmonic function around a pole

I have a radial harmonic function $h:\mathbb R^N\backslash\{0\}\to\mathbb R$ which has a pole of order $m$ in 0, and I would like to compute $$ \frac{1}{\sigma_N}\int_{\partial ...
0
votes
1answer
44 views

Finding the distance between a plane and $(0,0,0)$

Given the lines: $ \frac{x+1}{4} = \frac{y-3}{1} = \frac{z}{k} $ and $\frac{x-1}{3} = \frac{y+2}{-2} = \frac{z}{1} $ that lie on the same plane. How can I find the parameter $k$ ? (I guess ...
3
votes
2answers
60 views

Minimum value of the function $\sqrt{(1+1/m)(1+1/n)}$

If $m, n$ are positive real variables whose sum is a constant $k$, then what is the minimum value of $$\sqrt{\bigg(1 + \frac{1}{m}\bigg)\bigg(1 + \frac{1}{n}\bigg)}$$
0
votes
0answers
105 views

Limit of a piecewise function separated by functions

I have to decide whether the following function is continuous or not: $$\displaystyle f(x,y)=\begin{cases} \frac{(x-2y)\sin(xy)}{xy^2+y} & \text{if } y \neq 0 \wedge xy \neq -1\\ x^2 ...
1
vote
2answers
75 views

Level curve of $f(x,y) = cos(x+y)$

Find and sketch the level curve of the function $f(x,y) = cos(x+y)$ when $f(x,y) = -1$ I've tried to see how this curve is using implicit derivative, but I have no clue of how this curve is. Thank ...
0
votes
1answer
136 views

Given the following set, how do I determine the boundary, exterior and interior points?

$$A = \{(x,y) \in \mathbb R^2: 2 \leq x <5, 1 < y \leq 4\}$$ Is $A$ open? Why or why not? Is $A$ closed? Why or why not? Determine the boundary points of $A$. Determine the exterior points of ...
1
vote
1answer
54 views

$\lim_{(x,y) \to(0,0)}\sin(x - y)$

So I tried approaching from $x=0, y=0, y=x, y=x^2$, and $y=x^3$ and the resulting limits are all $0$ but apparently the limit doesn't exist. Why is this so?
0
votes
0answers
30 views

Multiple rate of change system equation

This is the complete and more complex version of the problem that I posted earlier at How to create a multiple rate of change equation The problem: Two bubbles in an airtight container are full of ...
0
votes
1answer
60 views

Prove that a plane is continuous?

Okay, so the problem gives a matrix: $$ \pmatrix{ 1&2\\ 3&4} $$ and this matrix is an $\Bbb R^2 \to \Bbb R^2$ linear map. I am asked to explicitly write the component functions of $A$, and ...
5
votes
1answer
150 views

Extended matrix function

I have a continuous matrix-valued function $f:\mathbb{R}^d\mapsto {\cal M}_{k\times d}$, with $d<k$, such that $f(x)$ is full rank for all $x\in\mathbb{R}^k$. Can I extend this function to be a ...
7
votes
2answers
359 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem. Let $f(x):\Omega\longrightarrow \mathbb R^n$ be a $C^2$ function where $\Omega$ is an open subset of ...
0
votes
1answer
62 views

Property of euclidean norm - proof

In my textbook there is lemma: $(\forall\vec{a} \in \mathbb{R}^{n})||\vec{x}||=0 \Leftrightarrow \vec{x}=0$ that should be easy to prove. And I have $0\le x_{i}^2\le ...
2
votes
2answers
335 views

Using a double integral to find a volume by revolving a region R around the y-axis.

The question asked is as follows: "Find the volume of the solid of revolution obtained by rotating the area bounded by the curves about the line indicated. $$y = |x^2 - 1|, x=-2, x=2, y=-1.$$ Rotate ...
0
votes
2answers
38 views

Double integral of a region.

Could someone help with the following question please: For shape one I think it is just $ \int_{-1}^{1} \mathrm \int_{-1}^{1} \mathrm{f(x,y)}\,\mathrm{d}xdy $
1
vote
1answer
35 views

How to find a derivative implicitly?

In class my professor gave u s the function $\sin{(xy)} = x^{2} + y^{2} + z^{2}$ and asked us to find $\frac{\partial x}{\partial y}.$ For this he said implicit differentiation was needed and wrote ...
2
votes
1answer
32 views

How to evaluate the double integral $\int _{0}^{1}\!\int _{{x}^{2}}^{1}\!{x}^{3}\sin \left( {y}^{3}\right) {dy}\,{dx}$?

This is an exam problem that should be solvable in less than 30 minutes: $$\int _{0}^{1}\!\int _{{x}^{2}}^{1}\!{x}^{3}\sin \left( {y}^{3}\right) {dy}\,{dx}$$ I have tried switching the order of ...
0
votes
2answers
21 views

Help understanding domains of two variables?

I am confused by this topic. Example: For the first one, I feel like I'd set the inner section to 0, because I know that you can't take the natural log of zero. I just am not sure how to correlate ...
1
vote
1answer
116 views

Why are there so many different definitions for differentiability?

I am studying differentiability for functions of several variables. Here is the first definition of differentiability I came across:$\quad$ A function $f:\Bbb R^n\to\Bbb R^m$ is differentiable at ...
1
vote
0answers
23 views

How to prove that $Df(a)(T_M(a))\subset T_N(f(a))$?

Let $M\subset\mathbb{R}^n$ and $N\subset\mathbb{R}^m$ be two manifolds and $f:U\rightarrow\mathbb{R}^m$ a function of class $C^1$ defined in an open set $U\supset M$ and such that $f(M)\subset N$. To ...
1
vote
1answer
39 views

Studying continuity of a function

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be given by $$ f(x,y) = \begin{cases} \frac{\sin(xy)}{\sqrt{x^2+y^2}}, & \text{if }(x,y)\text{ $\neq (0,0)$} \\ 0, & \text{if }(x,y)\text{ $= (0,0)$} ...
1
vote
0answers
37 views

Is that a valid way to prove indifferentiability?

Given the following exercise: Let $ f(x, y) = \begin{cases} \frac{x^3}{x^2 + y^2}, & \text{if $(x, y) \ne (0, 0)$} \\ 0, & \text{if (x, y) = (0, 0)} \\ \end{cases}$ Check whether ...
1
vote
1answer
39 views

continuity of a function from the plane to the line

Let $f$ be given as $$ f(x,y) = \begin{cases} \dfrac{ \sin x - \sin y }{x-y}, & \text{if }\text{ $x \neq y $} \\ \cos x, & \text{if } x \text{ $=y$} \end{cases} $$ My claim is that the ...
0
votes
2answers
32 views

Finding two variables at a time

Can we find out values of $a$ and $b$ in the equation $x^a - x^b = z$? Where $x$ and $z$ values are given. $x,z,a,b$ are positive integers. $a>b$. For example in the given equation $3^a - ...
1
vote
1answer
129 views

quasiconvexity of a function

Let $p>0$, $p\neq 1$. For $x=(x_1,\dots,x_n),x_i>0,\sum_ix_i=1$, $$f(x)=\sum_i (x_i/\|x\|_p)^{p-1}y_i,$$ where $y=(y_1,\dots,y_n),y_i>0,\sum_i y_i=1$ is fixed. Is $f$ quasiconvex? Plots ...
0
votes
1answer
400 views

The Derivative of a General Linear Map

This question is somewhat abstract compared to the things we've discussed in class, so I'm just making sure I've got the right idea. I'd appreciate any help/suggestions; I'm pretty sure I've got the ...
1
vote
2answers
45 views

Gradient Vector of a composite function

So I have a function $$r= ( x^2 + y^2)^{1/2}$$ and I want to show that $$\operatorname{grad} f(r) = f'(r)(\operatorname{grad} r).$$ I don't really know where to begin do you say that $f(r) = (f ...
1
vote
2answers
37 views

Global maxima of a function subject to a constraint.

I am trying to prove that the global maxima of $f(x_1,x_2,...,x_n)=(x_1x_2···x_n)^2$, subject to $\lVert(x_1,x_2,...,x_n)\rVert_2=r$ is $(r^2/n)^n$ I know I have to find the critical points of the ...
0
votes
1answer
248 views

How to find the curve of intersection of a ellipsoid and a plane?

Let $C$ be the curve of intersection of the ellipsoid $x^2+2y^2+3z^2=39$ and the plane $3x+y-7z=0$. Find the parametric equations for the tangent line to $C$ at $(5,-1,2)$. I don't know how to find ...
5
votes
4answers
170 views

How to find the minimum value of this function?

How to find the minimum value of $$\frac{x}{3y^2+3z^2+3yz+1}+\frac{y}{3x^2+3z^2+3xz+1}+\frac{z}{3x^2+3y^2+3xy+1}$$,where $x,y,z\geq 0$ and $x+y+z=1$. It seems to be hard if we use calculus methods. ...
0
votes
1answer
79 views

Please tell me how to evaluate this integral.

Compute the area of that portion of the conical surface $x^2 + y^2 = z^2$ which lies between the two planes $z = 0$ and $x + 2z = 3$. Ans:$2\pi\sqrt{6}$. Thank you. ...
1
vote
1answer
64 views

Maximum steepness of hill

A hill is given by $$ z = f(x,y) = \frac {32}{1 + x^2 + y^2}$$ where $z$ is the height of the hill in meters. At what height is the hill the steepest? The standard way to do it, which is how I ...
1
vote
1answer
34 views

Does this limit of arc length exist?

We have a parametrized curve $\gamma: \mathbb{R} \rightarrow \mathbb{R^2}$ given by $\gamma (t) = \langle e^t\cos (t), e^t\sin(t)\rangle$. I want to compute the arc-length of this curve on $[a,b]$ in ...
0
votes
1answer
34 views

Product rule type formula for $\nabla \cdot (M(x)v(x))$ where $M(x)$ is a matrix and $v(x)$ is a vector?

Let $M(x)$ be a $n\times n$ matrix with each element depending on $x$ a variable on $\mathbb{R}^n$. Let $v(x)$ be a vector. Is there a nice product rule formula for $\nabla \cdot (M(x)v(x))$?
1
vote
1answer
72 views

Change of variables in Double integral - explanation of solved example.

I'm exercising double integrals and here's an solved example that I'm not understand something. Calculate $\iint_D xy(x^2 + y^2) dx\,dy$ where $D$ in the first quadrant bounded by: $1 \le xy ...
0
votes
1answer
115 views

Jacobian matrix of an inverse diffeomorphism

Let $f:\Omega\rightarrow\mathbb{R}^3$ be a function defined as $f(x,y,z)=(x-xy, xy-xyz, xyz)$ and $\Omega=\{(x,y,z)\in\mathbb{R}^3:xy\neq0\}$ an open set. As f is an injective function in $\Omega$ ...
2
votes
0answers
37 views

Local extremes for quartic multivariable function

I have the function :$\ f(x,y)= x^4 +y^4-8xy $, and I have to find it's local extreme points. I computed the partial derivatives : $\ f_x= 4x^3-8y$ $\ f_y= 4y^3-8x $ $\ f_{xx}= 12x^2 $ $\ f_{yy}= ...
1
vote
1answer
70 views

proving existence of diffeomorphism

In my hand out of manifold, I found the following lemma but there is no proof there: Let $U\subseteq\mathbb{R}^m$ be open and pick some $a\in U$. Suppose that $f:U\mapsto \mathbb{R}^n$ is a smooth ...
0
votes
1answer
40 views

Restricted function

Let $A=\{(x,y): x,y\in(-1,1)\}$. Is there a function $f:A\mapsto A$ such that $f(x,0)=(x,x^2)$ $f$ differentiable and bijective on $A$. I have tried a lot of constructions but the problem is in ...
0
votes
1answer
43 views

If $\{x_p\}$ converges to $x$, then $\{x\}\cup \{x_p\}$ is compact

If $\{x_p\}$ converges to $x$, then $\{x\} \cup \{x_p\}$ is compact. My attempt: If $x_p$ converges to $x$, then any subsequence will also converge to $x$. Then the set $\{x\} \cup \{x_p\}$ has ...
0
votes
0answers
48 views

How to parametrise this surface integral

This is the question: $ S $ is the boundary of the region $ \{(x,y,z):0≤z≤h, a^2 ≤x^2+y^2 ≤b^2 \}$ where $ h,a,b$ are positive and $a<b$. ${\bf F(r) } = \exp(x^2+y^2){\bf r}$ where $ {\bf ...
2
votes
2answers
1k views

Shortest distance between two curves

Let $C_1= \{ (x, y) \in \mathrm{R}^2 : y = x^2 +1 \}$ and $C_2= \{ (x, y) \in \mathrm{R}^2 : x = y^2 +1 \}$, find the points which minimize distance between $C_1$ and $C_2$. What I tried is: we know ...
0
votes
2answers
38 views

Need some help with computing line integrals for vector fields

I am a little confused on the computation of a Line Integral of a Vector Field. Here is what I have so far: $$ \int_C \mathbf F \cdot d \vec r$$ (F is a vector field of n dimensions ($$ n \ge 2- ...
0
votes
1answer
45 views

Differentiability of linear least squares

Show that least-squares $\|y-X\beta\|^2$ is twice differentiable and has minimizer. I understand that the second derivative is $X'X$. Also it is a composition of linear function which is ...
0
votes
1answer
141 views

Triple integral over the region bounded by six planes

Evaluate $\displaystyle\iiint\limits_E \, \displaystyle\frac{\mathrm{d}V}{(x+y+z)^{3}}$, where E is the region bounded by the six planes $z=1$, $z = 2$, $y = 0$ , $y = z$ , $x = 0$ , $x = y + z$. ...
1
vote
0answers
42 views

Which are the level sets of this multivariable function?

I have to find the level sets of the function $$f(x, y) = \frac{1}{\sqrt{36 - 25x^{2} - 25y^{2}}},$$ so I make $f(x, y) = k$ and after manipulating this equation come to $$x^{2} + y^{2} = \left( 36 - ...
1
vote
1answer
22 views

polarcoordinate and just straight evaluation

Evaluate $\int_0^1 \int_0^1 xy(x^2+y^2)^{\frac{1}{2}}\ dy\ dx$. I started off using polar coordinate and then im stuck with defining the bound. And how would you solve it without polar coordinates
0
votes
2answers
49 views

How to calculate this multivariate limit changing to polar coordiantes?

I need to calculate the limit: $$ \lim_{(x, y) \rightarrow (0, 0)}{\ln{\left(\frac{19x^{2} -x^{2} y^{2} + 19y^{2}}{x^{2} + y^{2}}\right)}}.$$ Somebody suggested me to change to polar coordinates, so I ...
1
vote
2answers
481 views

Find the volume bounded by a cylinder.

Find the volume bounded by the cylinder $x^2 + y^2=1$ and the planes $y=z , x=0 ,z=0$ in the first octant. How do I go about doing this?
1
vote
1answer
41 views

Why is the partial derivative $f_x' = 0 $ is not continous?

Looking again at my first CalculusII exam and I get confused about something. Let $ f(x, y) = \begin{cases} (x^2 + y^2) \sin\left(\frac{1}{x^2 + y^2}\right), & \text{if $(x, y) \ne (0, 0)$} \\ ...
1
vote
4answers
131 views

Two-variable limit question

$\lim\limits_{(x,y)\rightarrow (0,0)} \dfrac{x^2y^2}{(x^2+y^4)\sqrt{x^2+y^2}}$ How to solve this two-variable limit? Thanks :D