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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32 views

Surface with every normal line passing through origin

We know that in $\mathbb{R}^3$, any line normal to a sphere passes through the origin. Is the converse true? Let $F(x,y,z) = 0$ be such a surface. Then, we have for some $t(x,y,z) \in \mathbb{R}$ $$ ...
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1answer
19 views

necessity of continuous partial derivatives?

In my old book in calculus, it says that a sufficient condition for the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ to be differentiable at an internal point z, is that the partial derivatives ...
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2answers
30 views

How to find the area through the given integral?

If $\int \int _Ax\:dxdy\:$, I want to find the area between the parabola $y=x^2$ and the straight line $2x-y+8=0$. I know that the equation of the straight line is $y=2x+8$, so after sketching the ...
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1answer
23 views

Making a function continuous

If we had $f(x,y) = \frac{g(x,y)}{h(x,y)}$ which is not defined for $(0,0)$ (as we divide by zero in this case and I show that the function $L(x) = \lim_{y \to 0} f(x,y)$ is not continuous... what ...
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2answers
31 views

Projection function preserving distance

Given $n>1$. I wis to construct a function $f:\mathbb{R}^n\mapsto\mathbb R$ such that $$\|X_1-X_2\|\le\|Y_1-Y_2\|\implies|f(X_1)-f(X_2)|\le |f(Y_1)-f(Y_2)|$$ for all $X_1,X_2,Y_1,Y_2\in\mathbb{R},$ ...
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1answer
46 views

What's wrong with my calculation?

I want to find the area of a triangle with vertices $\left(0,0\right)$, $\left(2,1\right)$, and $\left(2,0\right)$ with $\iint _A\left(2x-3y\right)dx \ dy$, i think the limits of the integral are ...
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1answer
43 views

Why does the order not matter? Partial D

When taking partial derivatives, why does the order not matter as long as the function is continuous? Any proof, intuitive or rigorous?
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0answers
72 views

Finding surface integral need help

Question is Evaluate double integral over S and integrand is given by xy / squareroot of 1 +2x^2 dS where the surface is S = { (x,y,x^2 +y)} : 0 less than and equal to x less than and equal to y , x+y ...
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1answer
10 views

length of intersection of parabolic cylinder and a surface

Let $C$ be the curve of intersection of the parabolic cylinder $x^2 = 2y$ and the surface $3z = xy$. Find the length of the part of $C$ from $(0, 0, 0)$ to $(6, 18, 36)$. (Hint: It may be useful to ...
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0answers
41 views

Why do we need partial derivatives?

Partial derivatives are used to find the slope of a three dimensional curve at an angle. But why do we need them? Can't another function be created from the first one which calculates $x$ and $y$ ...
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2answers
23 views

Proof of a property of directional derivative

I am stuck with the proof of the following proposition. I am given that the directional derivative of f exists at a with respect to the vector u, and I should prove that f'(a,cu)=cf'(a,u) I tried to ...
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0answers
21 views

Divergence of a radial $1/r$ vector field

Please explain how to obtain the divergence of the function $F(r,\varphi,\theta)=\hat{r}/r$. Is there a solution without computing the surface integral for definition of divergence? Thanks for your ...
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0answers
15 views

Multivariable calculus proof

The following is primarily from Loomis and Sternberg's Advanced Calculus. I have added some notes, marked between asterisks, e.g., * my notes * (these could be incorrect). Notation. $\Delta ...
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1answer
49 views

Find surface area using surface integrals

Find the area of surface of solid bounded by cone $z=3-(x^2+y^2)^{\frac{1}{2}}$ and paraboloid $z=1+x^2+y^2$ I have not been able to figure out figure of the question. Please help me with this.i ...
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0answers
16 views

Chain Rule for Multivariable Calculus: Dieterici's equation

A question came up in homework (2.5, q21, Vector Calculus 6th Edition Marsden, Tromba). It is an exercise in chain rule and I think it's asking me to use some implicit. Here's the question: ...
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0answers
31 views

The intuition of the rank theorem

In Rudin's Principle of Math Analysis, I know the proof of the rank theorem. However, I fail to understand its content. 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ ...
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2answers
41 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, ...
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0answers
50 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 ...
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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 ...
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1answer
28 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 ...
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0answers
20 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 ...
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1answer
22 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 ...
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1answer
39 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 ...
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1answer
15 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
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1answer
33 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
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1answer
53 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
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1answer
49 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
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1answer
15 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 ...
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2answers
14 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 ...
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0answers
10 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 ...
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2answers
22 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
25 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
17 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 ...
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2answers
15 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?
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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 ...
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1answer
47 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 ...
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1answer
24 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
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1answer
181 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 ...
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0answers
24 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} ...
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0answers
19 views

Finding Unit Vector

How do you find a unit vector that's parallel to both the planes 8x+y+z=1 and x-y-z=0? I've thought about using the cross product and dot product, but then I"m not entirely sure which one is the ...
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1answer
157 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 ...
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1answer
45 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 ...
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73 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?
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32 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 ...
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1answer
53 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 ...
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1answer
33 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 ...
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1answer
27 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 ...
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0answers
33 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 ...
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1answer
10 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 ...
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2answers
47 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 ...