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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0
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1answer
39 views

Work against friction is proportional to length of path

If, given that the frictional force is constant, one wants to show that the work done against friction is proportional to the length of the path, would this line of reasoning be correct? We can use ...
0
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1answer
45 views

Linear combination of gradient vector fields

Suppose we have that $\bf{F}=\vec{\nabla}$ ${f}$ and $\bf{G}=$ $\vec{\nabla}g$ are three gradient vector fields in $\mathbb{R}^n$, such that $\bf{H}=$ $c_1\bf{F}+$ $c_2\bf{G}$ for some $c_1, c_2 \in \...
3
votes
3answers
44 views

Given a function $f(x,y)$ find the limit as $(x,y)\to(0,0)$

If it exists, find $$\lim_{(t,x)\to(0,0)}\frac{t^2\sin^2(x)}{2x^2+t^2}$$ Along the curves $x=mt,t=0,x=at^2,t=ax^2$ the limit approaches 0; the graph also makes $L=0$ seem correct. So assuming that $...
0
votes
1answer
35 views

Partial derivatives and chain rule exercise

Let $f:\mathbb R^2\to \mathbb R$ be a function of class $C^2$ and $g:\mathbb R^2->\mathbb R$ the function given by $g(r,θ):f(r\cos(θ),r\sin(θ))$. If $f_x(1,1)=f_{yy}(1,1)=1$ and $f_{xx}(1,1)=f_y(1,...
0
votes
2answers
385 views

Does Green's Theorem hold for polar coordinates?

I was working on a proof of the formula for the area of a region $\mathcal R$ of the plane enclosed by a closed, simple, regular curve $\mathcal C$, where $\mathcal C$ is traced out by the function (...
0
votes
1answer
148 views

The divergence of gravitational potential

Is this computation of divergence correct? $$\mathbf{g} = \frac{Gm\mathbf{r}}{r^3}$$ $$\nabla\cdot\mathbf{g} = \frac{d}{dx}\mathbf{g}_x + \frac{d}{dy}\mathbf{g}_y + \frac{d}{dz}\mathbf{g}_z$$ $$r = ...
0
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1answer
25 views

Prove that the map $x:(u^1,u^2)\mapsto ((2+\cos u^1)\cos u^2,(2+\cos u^1)\sin u^2,\sin u^1)$ is injective on the domain $-\pi<u^1,u^2<\pi$.

So, like we usually prove injectivity, I consider my two points in $\mathbb{R}^2$ which I denote $(u^1,u^2)$ and $(v^1,v^2)$, and I'm trying to conclude that if $\underbrace{x(u^1,u^2)=x(v^1,v^2)}_{(*)...
0
votes
2answers
178 views

Critical values of $f(x, y) = x^3 − 147xy + 343y^3$

The only critical values I can find are $(0,0$) and $(7,1$). Apparently there is another one, and I can't find it which is very concerning. Just so we are clear, critical values are where $f_x = 0$ ...
1
vote
1answer
29 views

Integral over a sphere in spherical coordinates

Suppose we have a function $f(x,y,z)$ where $(x,y,z) \in S$. With $S$ being unit sphere in $R^3$. Passing to spherical coordinates we may write $x=\sin{\theta} \cos \phi$ and $y=\sin{\theta} \sin \phi$...
1
vote
1answer
14 views

Continuity and limit of union of curves

Let $f:\mathbb R^2 \rightarrow \mathbb R$ given by: $f(x,0)=1+x^2, \; f(0,y)=1+y^2 \text{ and } f(x,y)=0 \text{ if } x\neq 0 \text{ and } y \neq 0$ a) Is $f$ continuous on $(0,0)$? b) $\frac{ \...
2
votes
2answers
54 views

Intersection of two non-linear equations?

Graphically it is clear that $$1 + 2e^{{(x-y)}^2}(x-y) = 0$$ $$e^{{(x-y)}^2} - y = 0$$ has a unique solution, but how do I solve this analytically? If this is not possible, then what would be an ...
1
vote
1answer
53 views

Flux of a vector v through a surface S

This problem has myself and quite a few others stumped. Someone on another forum recommended this place, so I thought I'd give it a try. The question is this: Flux of a vector $v$ through a surface $...
0
votes
1answer
99 views

Deriving the COM of a hemispherical shell

Problem: Derive the COM of a hollow hemisphere of mass $M$ and radius $R$ using Iterated Integrals in Cylindrical Coordinates. I have no idea as to how to go about this problem using ...
3
votes
2answers
61 views

How to prove function $f(x,y)=\frac{1}{xy}$ is not uniformly continuous?

Here I consider uniform continuity of functions in $\mathbb{R}^n$. Take a function of two variables for example. We said that $f(x,y)$ is uniformly continuous if for any $\epsilon>0$, we can find ...
1
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2answers
42 views

Change of variable in the integral

Let $f:[0,b]\times[0,b]\rightarrow\mathbb{R}$ be continuous. Prove that $$\int_0^b dx\int_0^x f(x,y) \, dy=\int_0^b dy\int_y^b f(x,y) \, dx$$ La idea es usar los teoremas: \ Regla de Leibniz, de ...
0
votes
1answer
29 views

Gradient and tangent line

The gradient of $f(x,y) = x^2 + y^4$ is tangent to the curve $\gamma(t)=(t^2,t)$, at a point $P = \gamma(t_0)$, with $t_0 > 0$. Consider the level curve of $f$ that contains $P$. Find the equation ...
0
votes
1answer
41 views

Multivariable calculus.

I have just started multivariable calculus and we have been given a definition. $\text{We say that } l \in \mathbb{R} \text{ is a limit of a function } f \text{ at a point } P_0 \text{ if for every } ...
2
votes
1answer
24 views

Equivalent condition to differentiability of a function in a general set.

If $A\subset \mathbb{R}^k$ is an arbitrary set one said that $f:A \rightarrow \mathbb{R}^n$ is differentiable if for each point $x\in A$ exists an open set $U_x$ and a function $\tilde{f}:U_x \...
1
vote
1answer
34 views

When to type a function in bold?

Is it convention to bold any function with more than one output? For example, $\textbf{f}:\textbf{R}^2 \mapsto \textbf{R}^3$ or $f: \textbf{R}^2 \mapsto \textbf{R}^3$ $\textbf{f}(\textbf{x})$ or $f(\...
0
votes
2answers
29 views

What is wrong with my reasoning in this differentiation under the integral sign problem

Let $y(t)$, $f(t)$ be $C^2$-functions satisfying $y(x) = 4\int_0^x(t − x)y(t)dt−\int_0^x(t − x)f(t)dt.$ Show that $y$ solves the differential equation $y''(x) + 4y(x) = f(x)$, with the initial ...
0
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1answer
30 views

Can anyone explain these two inconsistent results? Partial derivative calculation.

Let $x=r\cos \theta$ and $y=r\sin \theta$. Find $r_x$. My answer: $r_x=(r_x)^{-1}=x_r=(\cos \theta)^{-1}$. Book answer: $$\frac{\partial (r^2)}{\partial x}=\frac{\partial (x^2+y^2)}{\partial x} \...
13
votes
0answers
458 views

Computing the volume of a region on the unit $n$-sphere

I would like to compute the surface volume of a region on the unit $n-1$-sphere: $$\sum_{i=1}^n x_i^2 = 1,$$ bounded by an ellipsoid $$\sum_{i=1}^n a_ix_i^2 \leq a_2,$$ where $1=a_1 < a_2 <...
0
votes
2answers
45 views

Find a function $g(x,y)$ that satisfies the following conditions

Find the function $g(x,y)$, that is continuous at $(0,0)$ but the partial derivatives at $(0,0)$ do not exist.
0
votes
0answers
27 views

Calcule $\dfrac{\partial b}{\partial x}$ where $b=b(x(t),y(t))$

Consider $$\begin{cases}x=x(t)=x_2+t(x_3-x_2)\\y=y(t)=y_2+t(y_3-y_2)\end{cases}\quad t\in[0,1] \tag{$*$}$$ and $b(t)=t(1-t)$, where $x_2,x_3,y_2$ and $y_3$ are known constants. Calculate $\dfrac{\...
4
votes
2answers
597 views

Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that $u \in C^2(\overline{B}_r)$ satises \begin{align} -\Delta u(x)=f(x), & \qquad x\...
0
votes
2answers
40 views

How to find this partial derivative using multivariable chain rule?

Suppose $$f(x,y,z)=x^3-y^2z^2$$ and also that $$x(u,v,w)=u+v,~y(u,v,w)=u-v,~z(u,v,w)=u+\sin{w}$$ if $$f(x,y,z)=f(x(u,v,w),y(u,v,w),z(u,v,w))=F(u,v,w)$$ using the chain rule find $$\frac{\partial f}{\...
0
votes
1answer
47 views

2-forms as wedge product of 1-forms

Why can all $2$-forms on $\operatorname{T}_p (\mathbb{R}^3)$ be written as product of $1$-forms? What is a counter-example to show this isn't true in a space other that $\operatorname{T}_p (\mathbb{...
1
vote
3answers
30 views

Find all $x,y$ for which $\nabla f$ forms an angle of $45°$ with the vector $(1,1)$

Let $f=x^2+y^2 \implies \nabla f=(2x,2y)$. Find all $x,y$ for which $\nabla f$ forms an angle of $45°$ with the vector $(1,1)$. So I thought of taking the dot product $2x+2y=\nabla f \cdot v=||\...
10
votes
6answers
350 views

Continuity of $f(x,y)=4x^3y^{11}(x^4+y^8)^{-2}$ at $(0,0)$

Well, the function is $$\frac{4x^3y^{11}}{(x^4+y^{8})^2}$$ and we want to know if it's continuous at $(0,0)$. I've tried as many trajectories as I could think of, and they all give $0$ as the limit. ...
0
votes
1answer
21 views

Type of extremum in Lagrange Multiplier Method

Let's say I'm given that there are rectangular boxes all of which have a constant surface area, say S. Now, I want to find the box with either the maximum volume or the minimum volume. If I apply ...
0
votes
2answers
626 views

The motion of a solid object

The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density at the point and occupies a region W, then the ...
0
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2answers
32 views

Finding boundaries of integration for 3-D Surface

I'm having a hard time: (1) visualizing the function $z = |x-y|$ (2) finding the limits of integration of $x$ and $y$ over the unit square. Do you guys have any tips to help me get started? I ...
1
vote
1answer
134 views

Nearest and farthest point from an ellipse to a line segment

Find the nearest and farthest point from the ellipse $ x^2 + 3y^2 =3 $ to the segment made by $ x+y = 3 $ in the first quadrant. Found in a multivariable calculus course. So I have to find the ...
2
votes
2answers
62 views

Why don't we consider $\mathbb{R}^3$ to be an affine space?

When we're introduced to $\mathbb{R}^3$ in multivariable calculus, we first think of it as a collection of points. Then we're taught that you can have these things called vectors, which are (...
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0answers
14 views

Proof that the tangent vector of an arc length parametrized curve has a magnitude of one

Is this an acceptable proof? For a vector valued function $f(s)$, where $s$ is arc length. $\frac{df}{ds} = \frac{df}{dt}\frac{dt}{ds} = f'(t)\frac{1}{\left|f'(t)\right|}$ Therefore $\left|\frac{df}...
0
votes
1answer
80 views

Rudin's application of the mean value theorem

I am studying theorem 6.26 (page 152) in Rudin's "Functional Analysis" that presents distributions as derivatives of continuous functions. Right at the beginning of the proof, if $\Omega$ is the ...
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3answers
51 views

Finding the area using double integrals.

We need to find the area of the shaded region , where curves are in polar forms as $r = 2 \sin\theta$ and $r=1$. I formulated the double integral as follows : We find the area in the first ...
3
votes
2answers
53 views

Multi-Variable Limit Problem

I am having difficulty showing that the limit of $$f(x,y) = \frac{x^2+y^2\sin(x)}{x+y}$$ as (x,y) goes to (0,0). does not exist. All the usual tricks of computing the limit along $(0,x)$ and $(0,y)$ ...
0
votes
1answer
24 views

dA in polar coordinates using total differentials

I'm trying to derive the area element $dA$ in polar coordinates using total differentials (just for the sake of trying: I know the existence of the Jacobian), but I can't get the correct result. ...
0
votes
1answer
71 views

The gradient is everywhere perpendicular to the contour lines of a function

In this article, there's something saying: The gradient is everywhere perpendicular to the contour lines of a function It justifies it by saying: Since along contour lines the change in ...
1
vote
2answers
78 views

Based on a coordinate system centered in a sphere where is $M(x, y, z) = 6x - y^2 + xz + 60$ smallest?

I am trying to work through a few examples in my workbook, and this one has me completely dumbfounded. Suppose I have a sphere of radius 6 metres, based on a coordinate system centred in that sphere,...
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0answers
32 views

Total derivative of function on a curve

Consider the function $$F=F(x,y,z)$$ where $x,y,z$ describe the curve $x=x(u)$, $y=y(u)$, $z=z(u)$ Since the total differential is $$dF(x,y,z)=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\...
1
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1answer
70 views
0
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2answers
255 views

Transform square region to triangular region

How do you express x and y in terms of u and v so that the region $\{(u,v): 0\le u, v\le 1\}$ is mapped to the triangular region in the $xy$-plane with vertices $(0,0)$, $(1,0)$, and $(0,1)$? Now, ...
0
votes
1answer
10 views

Centroid of bounded region.

Given a closed $n-1$ dimensional object: $x_1 \dots x_n = f_1(t) \dots f_n(t)$, what is the centroid of the $n$ dimensional region it encloses?
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0answers
25 views

Clarification on the definition of the complex line integral: do we always take the absolute value of the z'(t) factor in the integrand?

So far, when I have had to use the definition of the line integral, the z'(t) factor turns out to be positive, so it has not been an issue for me. But, some definitions that I read online take the ...
2
votes
0answers
33 views

Tangent unit vectors?

The question asks to find the unit tangent vector at the specified value of the parameter. I think I know the process, but I can't for the life of me figure out why I am getting the wrong answer. $r(...
0
votes
1answer
35 views

Using chain rule and differentiation under the integral sign

I'm working through Vector Calculus by Tromba & Marsden in preparation for my MGREs next month. I'm stuck on the following problem. Show that $\,d/dx \int_0^x f(x,y)\,dy = f(x,x) + \int_0^x \,df/...
3
votes
1answer
30 views

$f(\textbf{x}) = {a\over{|\textbf{x}|^{n-2}}} + b,\text{ }\textbf{x} \neq 0$

If $f(\textbf{x}) = g(r)$, $r = |\textbf{x}|$, and $n \ge 3$, show that$$\nabla^2 f = {{\partial^2 f}\over{\partial x_1^2}} + \dots + {{\partial^2f}\over{\partial x_n^2}} = {{n-1}\over{r}}g'(r) + g''(...