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
14 views

$f:\mathbb{R}^2\to\mathbb{R}^2, f(x,y)=(x+2y+y^2+|xy|,2x+y+x^2+|xy|)$

$f:\mathbb{R}^2\to\mathbb{R}^2, f(x,y)=(x+2y+y^2+|xy|,2x+y+x^2+|xy|)$, then $f$ is discontinuous at $(0,0)$ $f$ is continuous at $(0,0)$ but not differentiable at $(0,0)$ $f$ is differentiable at ...
2
votes
0answers
21 views

How to find that triple integral?

How to find the triple integral of $$ \frac{(z-z_0)z}{\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}}$$ over the sphere $ \{(x,y,z):x^2+y^2+z^2 \le 1 \} $ under the assumption $x_0^2+y_0^2+z_0^2 \le 1?$ Its ...
1
vote
0answers
16 views

Partial Derivatives [on hold]

What are the following partial derivatives, given: $$\small p=\frac{\exp\left(\frac{\rho-1}{2}\log(x^2+y^2)-\sigma\arctan\frac{y}{x}\right)}{(e^{-x}\cos y-1)^2+(e^{-x}\sin ...
0
votes
0answers
16 views

Expectation of a logarithmic/trigonometric function

I am trying to find a closed form solution of the following expectation: $$\mathbb{E}[\log(a+b\cos(\phi))]$$ where $a$ and $b$ are real constants, and the expectation is with respect to $\phi$. If ...
3
votes
1answer
50 views

Math GRE: Calculus Textbooks - is Spivak + Stewart + Rudin sufficient?

Recently, I splurged and spent $1000 in math textbooks in preparation for the Mathematics GRE subject test. So far, in terms of calculus books, I have purchased Spivak, Stewart, and Baby Rudin. Is ...
2
votes
5answers
114 views

Multivariate limit $\lim_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$

$$\lim_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$$ (a) Prove that the limit of $f(x, y)$ as $(x, y)$ approaches $(0, 0)$ along any straight line is $0$. (b) Does $\lim_{(x,y)\to(0,0)} ...
1
vote
1answer
54 views

Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
1
vote
0answers
26 views

On $f:A\to\mathbb{R}^2, f(x,y)=({x\over 1+x+y},{y\over 1+x+y})$

$A=\{(x,y)\in\mathbb{R}^2: x+y\neq -1\}$ $f:A\to\mathbb{R}^2, f(x,y)=({x\over 1+x+y},{y\over 1+x+y})$,Then Jacobian matrix of $f$ does not vanish on $A$ $f$ is infinitely differentiable on $A$ $f$ ...
2
votes
1answer
41 views

Relationship between Surface Area and Volume

Question: Is there a general relationship between surface area and volume analogous to the below examples? Example 1. Consider a ball $B$ centered at the origin of a spherical coordinate system. The ...
-2
votes
0answers
13 views

how to increace the volume to a specific volume in revolution of solid, using integration [on hold]

two functions, f(x)= 1/9(x-2)^2+7 domain range:{0,10}, g(x)=1/7(x-5)+0.7 domain range: {10,13} increase the volume to 1000ml to 1050mL using integration.
0
votes
2answers
32 views

Finding the partial derivatives of $V (x, y) = U (x, y)e^{−ax−by}$

I think I did something wrong, so I was hoping someone might be able to show me the solution Two functions $V (x, y)$ and $U (x, y)$ are connected by the equation $$V (x, y) = U (x, y)e^{−ax−by}$$ ...
3
votes
3answers
63 views

Changing order of integration (multiple integral)

Prove $$ \int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt $$ where $a$ is constant. So I began with two ...
2
votes
3answers
38 views

Finding the change of variables to transform $u_{tt} - u_{xx} = 0$ into $u_{rs} = 0$

I'm just beginning to introduce myself to partial differential equations and one of the first problems presented in the textbook I have literally no idea how to do. I think the author intended the ...
0
votes
1answer
43 views

Area of a surface defined by two functions

Find the area of the surface defined by $$x + y + z = 1$$ $$x^2 + 2y^2 \leq 1$$ I'm having trouble finding the parametrization of the surface
0
votes
1answer
30 views

Question about index notation on partial derivatives.

I've been studying quantum field theory a little bit and I've encountered a notation like the following: $$\mathcal{D}_{x,x'}=\frac{\partial}{\partial x^\mu}\frac{\partial}{\partial ...
1
vote
2answers
53 views

Evaluating the limits $\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$ and $\lim_{(x,y)\to(\infty,8)}(1+\frac{1}{3x})^\frac{x^2}{x+y}$

I got the following problem: Evaluate the following limits or show that it does not exist: $$\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$$ and ...
0
votes
1answer
41 views

Prove that if $\|f(x)\|\to 0$, then $f(x)\to 0$ using the $(\epsilon,\delta )$-definition of limit

Could anyone help me with this proof? Prove that $\lim_{x\to a} f(x)=0$ whenever $\lim_{x\to a} \|f(x)\|=0$ for $x\in \mathbb{R}^n$ Where $f(x)$ is a vector function and $\|f(x)\|$ is the ...
1
vote
1answer
43 views

how $3i \times 3i = 9i \times i$? (i is the unit vector and $\times$ is cross product)

$i$ is the unit vector; didn't know how to write it. I'm reading a text and somewhere it uses something like $ai \times bi = (ab)i \times i$ (implicitly). I can see why this is true geometrically, ...
0
votes
1answer
41 views

Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
0
votes
1answer
21 views

how many variables are there from 9 digits excluding repeat numbers

I have the numbers 1 to 9 I need to know how many different 9 digit code variations i would have using 1-9 but excluding any "next digit" replications. example: 123456789 is acceptable 112345678 ...
0
votes
1answer
17 views

parametric equations multivariate calculus

could anyone help me to solve this problem Given a parametrization of the tangent line to the curve,(x(t),y(t)) at t=a is: ...
0
votes
1answer
26 views

Triple integration, a general question

If the triple integral of the function g is equivalent to the triple integral of the function w, is it the case that g=w?
2
votes
1answer
28 views

Prove that $f$ is differentiable in $(0,0)$ if and only if $\lim_{t\to0+} g(t)$ exists

Let $g:[0,\infty)\to\mathbb{R}$ be a mapping and $f(x,y)=xg(\sqrt{x^2+y^2})$ for all $(x,y)\in\mathbb{R^2}$. Prove that $f$ is differentiable in $(0,0)$ $\iff$ $\lim_{t\to0+} g(t)$ exists. My ...
0
votes
1answer
30 views

Tractrix exercise

Exercise: Let $$\begin{align*}\gamma:(0,\pi) &\to \mathbb R^2\\ t &\mapsto \gamma(t)=(\sin t,\ \cos t+\log \left(\tan(\frac{t}{2}) \right), \end{align*}$$ be the parametrized curve of the ...
1
vote
1answer
15 views

Cycloid arc lenght question

I've parametrized the cycloid with the function $$\gamma(t)=(\cos(\frac{3}{2}\pi-t)+t,\sin(\frac{3}{2}\pi-t)+1) ; \space t \in [0,+\infty)$$ I am asked to find the arc lenght of the curve which ...
0
votes
2answers
37 views

Find $\iiint_E sin^3 x+\tan y+ 6\hspace{1mm} dV$, where $V$ is region inside $x^2+y^2+z^2 = 1$

I guess that the integral of $\sin^3 x+\tan x$ part is zero, because i have seen many problems like these where the integral is over a symmetrical region and the functions are odd. But I want ...
2
votes
2answers
74 views

Why is continuous differentiability required?

I have two questions. My book proves that if $f:\mathbb{C}\rightarrow \mathbb{C}$ is a holomorphic function, then it satisfies the Cauchy-Riemann equations, and if we look at the function as $F: ...
2
votes
3answers
36 views

Finding the partial derivatives of $h(x)=\int_{0}^{\|x\|} f(t)\, dt$

Find the partial derivatives of $$h(x_1,\dots,x_n)=\int_{0}^{\|x\|} f(t) dt$$ where $\|x\|$ is the Euclidean norm of $x=(x_1,\dots,x_n)$ and $f$ is some continuous function. I'm sorry but I'm really ...
3
votes
1answer
94 views

Finding multivariable limit

I would like to find the following limit $$\lim_{(x,y,z)\to(0,0,0)}\frac{x^3yz+xy^3z+xyz^3}{x^4+y^4+z^4}.$$ It looks like it would be zero since if we put $M=\max\{x,y,z\}$ and $m=\min\{x,y,z\}$, ...
1
vote
1answer
33 views

Constrained Optimization of a function of two variables.

I was given the following tutorial problem, and I'm having a bit of trouble seeing how it works. I've been asked to find the four critical points of this system, with two of these being degenerate ...
5
votes
0answers
52 views

Symbolic manipulation inside integral

I'm an undergrad who has just completed the standard calculus sequence (1, 2, and multivariable). I've done well in the courses, however, things like the following, which is a derivation of kinetic ...
3
votes
2answers
94 views

$\Delta \vec{v}=0$ implies $\nabla\cdot \vec{v}=\nabla\times \vec{v}=0$?

\begin{align} \Delta\overrightarrow{v}&=\nabla(\nabla\cdot\overrightarrow{v})-\nabla\times(\nabla\times\overrightarrow{v})\\ ...
1
vote
1answer
32 views

Prove that the maximizing point configuration on the unit circle for a Vandermonde like functional is a picket fence

For $\lambda_i \in S^1 \subset \mathbb{C}$, consider the functional $H(\{\lambda_1, \ldots, \lambda_n\}):= \sum_{j < k} | \lambda_j - \lambda_k | $. I want to show that $H$ is globally maximized by ...
0
votes
0answers
23 views

When does a polynomial have finitely many critical points on a level set of another polynomial?

Suppose I have two polynomial functions $f$ and $g$ and I am interested in the critical points that $f$ has on a level set of $g$, i.e. $\{x\in \mathbb R^n : g(x)=a_1\}$ for some $a_1\in \mathbb R$ . ...
1
vote
2answers
42 views

Multivariable limit of rational function

Does the following limit exist? $$\lim_{(x,y) \to (0,0)}\frac{x^2y^3}{x^4+(x^2+y^3)^2}$$ I tried to solve this problem using polar coordinates, but I can't simplify it. I tried the squeeze theorem, ...
1
vote
1answer
29 views

Finding the critical points in a constrained optimization problem using the Lagrangian

I've been given the following constrained optimization problem, but I'm having trouble even getting the critical points out - the numbers just seem way too complicated... Find the local maxima and ...
2
votes
0answers
73 views

Limits for multivariate functions $\displaystyle \lim_{(x,y) \to (0,0)}\frac{xy^2}{x^2+y^4}$ [duplicate]

Can anyone help me solve this question Does the limit exists? $$\lim_{(x,y) \to (0,0)}\frac{xy^2}{x^2+y^4}$$ I tried using polar coordinates to solve this problem but I got $\frac{0}{0}$ which is ...
2
votes
2answers
100 views

A continuously differentiable function with vanishing determinant is non-injective?

(This question relates to my incomplete answer at http://math.stackexchange.com/a/892212/168832.) Is the following true (for all n)? "If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously ...
1
vote
0answers
59 views

Verification: Hessian of the following composition.

I was hoping that someone could verify the steps of computing a Hessian matrix. I have the following function, $F:\mathbb{R}^n\to\mathbb{R}$, $$F({\bf x}) = \sum_{i=1}^mf(g(A_i^T{\bf x}))$$ where ...
1
vote
0answers
33 views

Applying Green's formula

I've been having some trouble proving this, so any help or suggestions would be great. Thanks in advance! Show using a Green's formula that, for any $u \in H^2(\Omega)$ satisfying $$u = ...
0
votes
1answer
33 views

Using the implicit function theorem to solve for two of four variables in the system of two equations

Show that there are positive numbers $p$ and $q$ and unique functions $u$ and $v$ from the interval $(-1-p, -1+p)$ into the interval $(1-q, 1+q)$ satisfying $$xe^{u(x)} +u(x)e^{v(x)}=0=xe^{v(x)} ...
2
votes
0answers
34 views

Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
1
vote
2answers
33 views

finding multivariable limits for a function

Could anyone help me with this Find $$\lim_{(x,y) \to (0,0)} \frac{3x^2y}{x^2+y^2}$$ if this limit exists I tried using the squeeze theorem, but i could not find a suitable expression for the ...
0
votes
0answers
18 views

Converting a slope field into a vector field

I have homework on slope fields where I have to graph a bunch and find the equillibrium solution, but instead of taking such a long time to graph them, I decided to use WolframAlpha. Sadly, there is ...
2
votes
2answers
28 views

Extract a variable from a formula

my maths a a bit rusty and I need to extract a variable from a formula. It's needed for a project about air quality in order to convert data from sensors to an index. The formula is : $$\left ...
2
votes
0answers
72 views

Maximize $\sqrt{x^2+y^2+z^2}$ subject to several conditions on $x,y,z$. [closed]

Find the maximum value of the following expression, $$R(x,y,z) = \sqrt{x^2 + y^2 +z^2} $$ if we are given that: $$\begin{align} y+4^{y+z} & \leq x + 4^{x+z} \\\ x^2 + y^2+8x & = 0 \\ 4 ...
0
votes
3answers
32 views

Continuity of multivariate functions

Determine whether the function $$ f(x,y)=\left\{\begin{array}{@{}l@{}} (x^3-y^3)/(x^2+y^2) & \text{for }(x,y)\neq(0,0)\\ 0 & \text{for }(x,y)=(0,0)\end{array}\right. $$ is continuous. To do ...
1
vote
1answer
19 views

Stereographic projections - equation of a plane question

The proof I'm trying to understand. I don't get why $k$ is unique. When trying to find the equation of a plane, suppose we're given a normal vector $n=(x_o,y_o,z_o)$ and a point on the plane ...
0
votes
1answer
47 views

Integral of $f$ over all space is $0$ implies $f=0$?

If you have that the integral of a function in all space (in my particular case is three dimensional space) is zero, under what conditions can you say that the argument is null over space? What ...
1
vote
1answer
38 views

To show unique solution for the Laplace equation

The problem is in the top while its weak form at the end, source $\hskip 1in$ I know that the solution is unique because the boundary condition is Dirichlet. But I want to show this. How can you ...