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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2answers
47 views

How can I solve these two tough integrals?

\begin{equation*} J_{1} = \int_{0}^{\sqrt{{\pi}/{6}}} \int_{y}^{\sqrt{{\pi}/{6}}} \cos{(x^2)}\,dx\,dy \end{equation*} \begin{equation*} J_{2} = \int\int_{E}\int z e^{(x^2+y^2)} + xe^{x^8}\,dV, ...
2
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0answers
64 views

Book Request: Taylor's Theorem for functions $f: \Bbb R^n \to \Bbb R^m$

I'm looking for a resource (e.g. a book, website, or arxiv paper) that goes over the general case of Taylor's theorem, with a full proof and examples. Do you guys know of any material that covers ...
1
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2answers
50 views

How do I solve that limit?

I've just started solving 2-variable limits and I'm stuck at one of the examples: $$\lim_{(x,y)\to(0,0)} \frac{1-\cos(x^2+y^2)}{x^2y^2(x^2+y^2)}$$ How do I approach limits like that? I've been ...
1
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1answer
37 views

How do you call this kind of functions in english?

I have a couple of formulas that I would like to plot, but I can't find the much needed documentation for them because I don't know how to correctly name them in english . This formulas assume that ...
1
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0answers
25 views

How to take the second order partial derivative

Given two functions $f=f(x)$ and $u=u(x,y,z)$, where $x,y,z$ are independent, how do I get the second order derivative $\partial^2f/\partial u^2$? My attempt: $$\frac{\partial^2 f}{\partial ...
0
votes
1answer
26 views

find out the volume of solid removed?

i have sphere that has an equation $$x^2+y^2+z^2=1$$ a cylindrical hole $x^2+(y-1/2)^2$=$1/4$ is cut through it . find the volume of the portion cut. i don't know what to do, i was thinking of using ...
1
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1answer
52 views

Double Integral Calculation

I am confused as to how the red arrow step was preformed. If I type the same integral into Maple I get $1-e^{-x}-e^{-y}+e^{-x-y}$ which is the same that I manually calculated, clearly not the same ...
0
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1answer
29 views

Relation between directional derivatives and derivative? [duplicate]

Is it possible to say that the directional derivatives of a function f at a exists but f is not differentiable at a? If so, why? I cannot get the intuition about it. Could someone please elaborate on ...
2
votes
1answer
67 views

Equation of ellipsoid surface obtained by revolving an ellipse

I'm working through the following example from the Princeton Review book: If the ellipse $x^{2} + x^{2/9}=1$ in the $xz-$plane is revolved around the $z-$axis, what's the equation of the resulting ...
2
votes
1answer
80 views

Apostol vector calculus exercise

I am self-studying multivariable calculus using MIT's publicly available materials, and I have been stumped by this exercise from Chapter 14.4 of the first volume of Apostol's calculus text: A ...
2
votes
1answer
63 views

Derivative of Quadratic Form as a Linear Approximation

I'm trying to find the derivative of the $quadratic$ form, for a $symmetric$ $n$ by $n$ matrix A and $ x \in \mathbb{R}^n $, $$ f(x) = x^tAx $$ such that the derivative is a linear map from $ ...
1
vote
1answer
19 views

Calculate average motion of points in a sphere

I have a sphere full of individual particles. Each particle has an $(x,y,z)$ co-ordinate and velocity in $(v_x, v_y, v_z)$ directions. I want to find out if there is any preferred direction of motion ...
1
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2answers
19 views

Expressing a unit tangent vector in terms of r(t)

Is there a simple way to express $N(t)$, the unit normal vector of a vector curve, in terms of $r(t)$? I know that $T(t)$=$\frac{r'(t)}{||r'(t)}||$ and that $N(t)$=$\frac{T'(t)}{||T'(t)||}$. Is it ...
0
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0answers
33 views

Making a function continuous (Multivariable Calculus) confirmation.

Can the function $$f(x, y) = 2xy / (x^2+2y^2)$$ be made continuous at (0, 0) by a judicious choice of the value f(0, 0) ? My answer is "No", for the reason that the limit DNE, as it can be ...
1
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2answers
32 views

Simple Chain Rule for Partials

This seems like a simple chain-rule question, but I'm getting stumped. I've searched and searched, but apologies if this question was covered somewhere else. ...
1
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1answer
30 views

what is the problem with this variable transformation?

$$\iint\limits_D (x − y)^2 \sin^2(x + y) \, dx \, dy$$ where $D$ is a parallelogram with vertices at $(π, 0), (2π, π), (π, 2π)$ and $(0, π)$. We can change the variables as $$x=\frac{u-v}{2} \text{ ...
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0answers
19 views

Need help with Change of variables.

I have an expression something like this $$\int_{\Omega_t} |\nabla_yu|^2 dy $$ and i want to change the variables via $y : \Omega \to \Omega_t $ $y= x+ tv(x)+ \frac{t^2}{2} w(x)$ Can someone help ...
1
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2answers
21 views

3 space Object as x gets large

I'm having a bit of trouble visualizing an object given $(a \cdot \cos(t), a \cdot \sin(t), ct)$ where c and a are constants. What object is described as c becomes large compared to a?
9
votes
2answers
1k views

Why not use two vectors to define a plane instead of a point and a normal vector?

In Multivariable calculus, it seems planes are defined by a point and a vector normal to the plane. That makes sense, but whereas a single vector clearly isn't enough to define a plane, aren't two ...
0
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1answer
62 views

How may I use this C loop to solve that integral?

Let C be the curve of polar equation $r = 2cos^2(\theta)$ and D the area bounded by the loop C which is situated in the half-plane $x \ge 0$ region. How may I calculate the D's area and use it to ...
1
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1answer
123 views

How to describe the region inside a sphere and below a cone in cylindrical and spherical coordinates?

If E is the region of space located inside the sphere $x^2 + y^2 + z^2 = 4$ and below the cone $z = \sqrt{3x^2 + 3y^2}$ How may I describe E in cylindrical and spherical coordinates? And how may I ...
2
votes
1answer
191 views

What is the easiest way to evaluate this integral?

\begin{equation*} \int_{0}^{64}\int_{\frac{1}{2}\sqrt[3]{y}}^{2} \frac{y^2}{\sqrt{x^{10} +1}} dxdy \end{equation*} I'm probably doing something really wrong, because I'm stuck. Any help will be ...
2
votes
0answers
108 views

Maximize and minimize a function using Lagrange multipliers.

I want to maximize and minimize $$h(a,b) = a + b$$ given the constraint $$g(a,b) = a^{\frac{1}{3}}b^{\frac{2}{3}} = l$$ I'm trying to use Lagrange multipliers. Here's what I did: \begin{align} ...
4
votes
1answer
216 views

calculation of normal derivative

Suppose $\Omega$ is a bounded region in the plane $\mathbb{R}^2$ with smooth boundary $\partial\Omega$. Suppose $u$ is a smooth function in $\Omega$. I want to calculate ...
0
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1answer
59 views

Prove that the range of $f$ is not closed.

I am having trouble computing the range of a function $f : \mathbb{R} \rightarrow \mathbb{R}^2$. My thinking would be that you just find the range of $\frac{2x}{x^2 +1}$ and the range of $\frac{x^2 ...
8
votes
1answer
123 views

If the second derivatives $f_{xx}$ and $f_{yy}$ exist, does $f_{xy}$ exist?

If the second derivative with respect to to $x$ exists ($f_{xx}$) and the second derivative with respect to $y$ ($f_{yy}$), does it follow that $f_{xy}$ exists?
1
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0answers
55 views

Prove that function is differentiable at $0$ if and only if $a>3/2$

Let $a>0$. I have to show that function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ $$f(x,y):=\frac{x^{2a}+y^{2a}}{x^2+y^2}$$ when $(x,y)\ne(0,0)$ and $f(0,0):=(0,0)$ is differentiable at $(0,0)$ if ...
1
vote
1answer
63 views

Computing the limit of a 2-variable function

Show that $\lim \limits _{(x,y)\rightarrow (0,0)} \frac{x^3y}{x^2+y^4}=0$ just using $\epsilon-\delta$ creterion. In fact, choose $\epsilon >0$ arbitary, then we have to find $\delta >0$ such ...
1
vote
1answer
55 views

Directional derivative of a function and continuity

I am studying directional derivatives, and I am stuck with how to visualize them. What it means geometrically to take the derivative of a function f at a point a relative to the vector u? My own ...
0
votes
1answer
28 views

Second partial derivative of $f(ax+by)$

I stumbled across this result: $$f_{xx}(ax+by) = \frac{a}{b} f_{xy}(ax+by)$$ Which I can't off the top of my head justify... I'm sure it's a very simple property but I can't seem to be able to ...
0
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0answers
38 views

hint for investigating positivity/negativity of a function

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function ...
0
votes
1answer
15 views

Simple composition of linear maps

Let $F : R^2 → R^2$ s.t. $x → (−x_2, x_1)$ and $G: R^2 → R^2$ s.t $x → (x_2, sin x_1)$. Evaluate $G ◦ F$ and $F ◦ G$. I have said $(G ◦ F)(x) = G(F(x))=-x_2,sinx_1$, but I feel as though this is ...
1
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2answers
141 views

Minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$.

I want to minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$, and I want to find the values of $a, b,$ and $\lambda$. This is what I've ...
1
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0answers
47 views

Minimizing a function using a direct approach (no Lagrange multipliers).

I want to minimize $g = x^2 + y^2$. My constraint is $h = 2x +y = l$. I know that using Lagrange multipliers is unnecessary here. I solved the constraint to get $y = l - 2x$. I then substituted this ...
5
votes
2answers
80 views

How to prove set $S=\{(x,y)\in \mathbb{R}^2~\vert~y>x^2\}$ is open (I need some hints)

Q: Prove $S=\{(x,y)\in \mathbb{R}^2~\vert~y>x^2\}$ is in open in $\mathbb{R}^2$. One of my intuitions: $S=\{(x,y)\in \mathbb{R}^2~\vert~y>x^2\}=\{(x,y)\in \mathbb{R}^2~\vert~y-x^2>0\}$ ...
0
votes
1answer
38 views

Having trouble proving this integral is infinite

I am working on an assignment and I have to prove that $$\frac{x^2-y^2}{(x^2+y^2)^2} \notin L_1\mathbb{R}^2)$$ to justify why Fubini's Theorem does not apply. I figured that the best way to do this ...
0
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0answers
44 views

Another versions of squeeze theorem

I have two questions regarding squeeze theorem: 1.Can I use squeeze theorem in higher dimensions? (I'm almost sure yes) If we have the strict inequality: $x_n\lt z_n\lt y_n$ when $\lim x_n=\lim ...
0
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1answer
30 views

Multivariable calculus, inner products

I am trying to solve this question. I have considered ith component and replaced it with $v_i/(v_i^2)^{1/2}$ and the summation form of the dot product, but cannot see how the RHS falls out, can ...
0
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0answers
45 views

(is my answer correct?) total differential of the function $u=\int _0^xe^{-s^2}ds\:$

if $u=\int _0^xe^{-s^2}ds\:$ i want to find $\frac{dx}{du}$ if i solve this through $\frac{d}{dx}\int _0^xe^{-s^2}ds=f\left(x,v\right)\frac{dv}{dx}-f\left(x,u\right)\frac{du}{dx}+\int ...
1
vote
1answer
59 views

general mean value theorem

Can anyone give me the intuitive explanation of the general mean value theorem stated in my notes as under: Let $f:U\rightarrow \mathbb R$ and $U\subseteq \mathbb R^n$ and let $f$ is differentiable ...
0
votes
1answer
32 views

continuity of $f(x,y)$ in $\mathbb R^2$.

I have to prove that the function given below is continuous in $\mathbb R^2$: $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$f(x,y)$= \begin{cases} e^{-\text(\frac{1}{|x-y|})} & \text{if ...
2
votes
1answer
72 views

How to prove it has a $\chi^{2}$ distribution

I tried to make $T$ close to $$T_1=\left(\frac{(x_1 - \mu_1)^2}{\sigma_1^2}+\frac{(x_2 - \mu_2)^2}{\sigma_2^2}-2 \frac{\rho}{\sigma_1 \sigma_2} \frac{x_1 - \mu_1}{\sigma_1}\frac{x_2 - ...
2
votes
2answers
28 views

Evaluating the triple integral $\int_{0}^{\sqrt{2}}\left(\int_{0}^{\sqrt{2-x^2}}\left(\int_{x^2+y^2}^{2}xdz\right)dy\right)dx$

I am trying to solve the following problem: Evaluate $$\int_{0}^{\sqrt{2}}\left(\int_{0}^{\sqrt{2-x^2}}\left(\int_{x^2+y^2}^{2}xdz\right)dy\right)dx$$ Sketch the region of integration and evaluate ...
0
votes
1answer
43 views

Finding the volume of a helical structure generated by the rotation of a square about an axis

A fixed line $L$ in 3-space and a square of side $r$ in a plane perpendicular to $L$ are given. One vertex of the square is on $L$. As this vertex moves a distance $h$ along $L$, the square turns ...
2
votes
2answers
61 views

Not exactly Partial Derivative

I've been just introduced to concept of Partial Derivative, My question is for some continuous and differentiable $g(x)$ we have $$g'(x)=\lim\limits_{\delta x \to 0} \left( \frac{g(x+\delta x) - ...
2
votes
1answer
115 views

Volume of sphere with triple integral

Using the same notations as in this picture : The element of volume is: $r^2 \sin(\theta) \, dr \, d\theta \, d\phi$ If I try to create the volume visually, I begin with integrating $r$ between ...
1
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1answer
57 views

Chain Rule for Second Partial Derivatives

I am trying to understand this: Let $g:\mathbb{R}\rightarrow\mathbb{R}$ and $f(r,s) = g(r^2s)$, where $r=r(x,y) = x^2 + y^2$ and $s = s(x) = 3/x$. What is (with Chain Rule) $$ ...
0
votes
1answer
95 views

The partial derivatives of the function $x=\int _u^ve^{-t^2}dt\:$ and $y=u^v$

If $x=\int _u^ve^{-t^2}dt\:$ and $y=u^v$, how to find $\left(\frac{∂u}{∂x}\right)_y$, $\left(\frac{∂u}{∂y}\right)_x$, and $\left(\frac{∂y}{∂x}\right)_u$ at $u=2$ and $v=0$? ...
0
votes
1answer
90 views

Vector Cross product - Rearranging issue

Given Data in question I have following relations in vector space$\begin{eqnarray}n_0^{'}(s)=-\kappa(s) \times n_0(s)\\n_1^{'}(s)=-\kappa(s) \times n_1(s)\\n_2^{'}(s)=-\kappa(s) \times ...
1
vote
1answer
269 views

Express partial derivatives of second order (and the Laplacian) in polar coordinates

$z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$ Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$ I'm having big troubles with using chain rule, in particularly the second ...