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
27 views

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuously differentiable function such that $f'(0)=0$. Define $g(x,y)=f(\sqrt{x^2+y^2})$

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuously differentiable function such that $f'(0)=0$. Define $g(x,y)=f(\sqrt{x^2+y^2})$. Is $g$ differentiable on $\mathbb R^2$? Any hints on how to do ...
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1answer
33 views

derivative of a function from $\mathbb R^n$ to $\mathbb R$

Let $ L:\mathbb R^n\to\Bbb R$ be the function $L(x)=\langle x,y\rangle$ , where $\langle,\rangle$ denotes some inner product on $\Bbb R^n$ and y is a fixed vector in $\Bbb R^n$. Further denote by DL, ...
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1answer
25 views

Partial derivatives as a basis?

Back when I was studying multivariable calculus, I had a sort of "intuition" for the formula for taking directional derivatives (the scalar product of the gradient and the vector that gives the ...
5
votes
3answers
50 views

Example of a Countable set which has volume zero.

The following question was asked in my exam which states: Give an example of a countable set which has volume zero. Hint: show that if $S=\{a_1,a_2,\ldots,a_n,\ldots\}$ is the set of points ...
0
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0answers
25 views

To evaluate a line integral

I need to evaluate line integral where C is a part for which $z\gt0$ and $z=0$ of intersection of surfaces $(x-1)^{2} + y^{2} = 1$ and $x^{2} + y^{2} + z^{2} = 4x$ . I am not writing about vector ...
0
votes
2answers
48 views

Surjection of the map $f:\mathbb R^{2} \to \mathbb R^{2}$

Let $f:\mathbb R^{2} \to \mathbb R^{2}$ be given by $f(x,y)=(x+y,xy)$.Then, which are correct? $f$ is surjrctive. The inverse image of each point in $\mathbb R^{2}$ under $f$ has atmost two ...
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1answer
14 views

Circulation using Green's Theorem

I have a problem where I'm asked to find the circulation along boundary of the half annulus with bounds $\{(r,\theta)|1\le r\le 3, 0\le \theta \le \pi\}$ in the force field $F =\ <-y, x>$. I ...
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2answers
48 views

Find linearization $L(x,y)$ of $f(x,y)$ at $(2,5)$ [closed]

$f(x,y) = 6 + x\ln (xy-7)$ My answer involves ln's but the answer to this question doesn't.
2
votes
1answer
34 views

Find the Area Using Polar Coordinates and a Double Integral

Of the area inside the smaller loop of the equation $r = 1-2sin\theta$ Here's my attempt at a solution: The shape has an inner and an outer loop, both of which will terminate at the origin. ...
1
vote
1answer
13 views

Find all the critical points that the function

$f(x,y) = 2xe^x\sin y$ has if $0 \leq y \leq 2\pi$ I found that the critical points are $(0,0)$, $(0,\pi)$, $(0,2\pi)$, $(-1, \frac{\pi}{2})$, $(-1, \frac{3\pi}{2})$. I lost points because I also ...
2
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2answers
53 views

Inconsistent answers when implicitly differentiating polar identities

Currently doing a problem where I need to find $\frac {\partial \theta}{\partial x}$. However, for $\tan(\theta)= \dfrac yx$, $\frac {\partial\theta}{\partial x}$ is yielding $- ...
1
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1answer
48 views

Optimization of a function

I need to optimize $$f(x,y,z)= x^2-y+e^{z}$$ with the restriction $$(x-2)^2+(y-3)^2+z^2=1$$ I've tried to substitute the restriction in $f(x,y,z)$ but it seems not to work. And when trying to use the ...
3
votes
1answer
81 views

differential forms, cylindrical coordinates, geometric interpretation [closed]

Consider a differential $1$-form $\beta$ which in cylindrical coordinates $(r, \theta, z)$ has the form $$\beta = f(r)\,dz + g(r)\,d\theta,$$where $g'(0) = 0$. Find a condition when $\beta \wedge ...
0
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0answers
12 views

When to use Stoke's over Divergence and vice versa

I'm still very confused as to when I'd use Stoke's Theorem over Divergence Theorem and vice versa. What are some very obvious things I could look for in a problem to determine which theorem to use? ...
0
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0answers
19 views

Volume of Parallelepiped

Show that the volume of the parallelepiped spanned by vectors $v_1,\ldots,v_n$ in $\mathbb R^n$ is $\det(v_1|\ldots|v_n)$. The parallelepiped is $A=\{a_1v_1+\cdots+a_nv_n:0\leq a_i<1 \}$ The ...
11
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2answers
211 views

quadratic form corresponding to function at critical point is positive definite implies local minimum

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^3$ function. Have $x_0$ be a critical point of $f$. How would I go about proving that if the quadratic form $q(h)$ corresponding to $f$ at $x_0$ is ...
1
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0answers
15 views

Evaluating an integral using Stokes' theorem

The Question Given a curve using the edges of the plane in the first octant $z=1-x-\frac17y$, evaluate $$\oint_C\textbf F\cdot d\textbf r$$ where $\textbf F=<e^{-x},e^x,e^z>$. My Attempt ...
0
votes
0answers
12 views

Computing the area bounded by cubic curves in the first quadrant

Let $a > b > 0$, $\alpha > \beta > 0$. I am asked to compute the area in the first quadrant bounded by the curves $x^3 = a^2 y$, $x^3 = b^2 y$, $y^3 = \alpha^2 x$, $y^3 = \beta^2 x$. What ...
2
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0answers
24 views

Given $S = \{ (x,y,z) : a \le x \le y \le z \le b\}$, finding $\int_S f(x) f(y) f(z) \, dx \, dy \, dz$

I am asked, given $S = \{ (x,y,z) : a \le x \le y \le z \le b\}$, to show that $$\int_S f(x) f(y) f(z) \, dx \, dy \, dz = \frac{1}{6} \left( \int_a^b f \right)^3.$$ What is a good way to approach ...
0
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0answers
20 views

One-form on R^n?

A $C^1$ one-form w defined on an open set U of any $\mathbb R^n$ is called closed if dw=0 on all U. 1- Show that w = $\frac {-y}{x^2+y^2}$dx+$\frac {x}{x^2+y^2}$dy is closed. 2- Calculate ...
0
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1answer
29 views

Is the gradient vector of a function the derivative of the function

Is the gradient vector the derivative of any function? I was wondering this because this works with a function of one variable.
0
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2answers
37 views

Showing $\frac{x^3y}{x^4+y^2}$ is continuous at all points

The function in the title is also defined to be 0 at (0,0). So the functon is obviously continuous at all points except (0,0), so I need to show that the limit of the function as (x,y) goes to (0,0) ...
0
votes
1answer
17 views

Volume of a beam using double integrals

I have a double integral of $1dxdy$, where the area is given by $x=[-1,3]$ and $y=[-4,1]$ I can just evaluate the integral by inspection, but I was wondering how to calculate this more properly. I ...
1
vote
1answer
21 views

Evaluate a line integral using the fundamental theorem of line integrals

Consider the vector field $${\mathbf F}(x,y)=(e^x)(\sin y){\mathbf i}+(e^x)(\cos y){\mathbf j},$$ and the curve $C$ composed of the graph of $\sqrt{x}+\sqrt{y}=5$ followed by segment from $(25,0)$ to ...
0
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1answer
18 views

Questions about definitions of tangent space

If we let S be a unit sphere, S = {P in R^3 : ||P|| = 1}. There are at least 2 definitions for the tangent space to S at a point P of S. The set of vectors perpendicular to the "radius vector" to ...
0
votes
1answer
33 views

Proof of vector identity

I have a problem proofing following vector identity: $$ \nabla \times (\vec a \times \vec b) = (\vec b \cdot \nabla) \vec a + \vec a (\nabla\cdot\vec b)-(\vec a \cdot \nabla)\vec b-\vec b(\nabla ...
0
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1answer
28 views

If $z=(x+y)f(ax+by)$ show that $(b(\partial z/ \partial x)-a(\partial z/ \partial y))(x-y)=(b-a)z$

If $z=(x+y)f(ax+by)$ show that $(b(\partial z/ \partial x)-a(\partial z/ \partial y))(x-y)=(b-a)z$. I don't really know where to start. I have written the total derivative for $f$ and $z$ assuming ...
1
vote
1answer
24 views

Taking the Derivative: Power Rule with Respect to Vector

I'm trying to take the derivative of \begin{equation} \phi\left(\mathbf{x}\mathbf{\theta}\right)\mathbf{x}^{\top} ...
0
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0answers
37 views

determining point on unit circle where a rat should be placed to get it started as fast as possible

For PDF of problem+ work, visit https://www.dropbox.com/s/ajah3bn9mwotha4/HW8P3.pdf?dl=0 Problem + proposed solution in images. My primary concerns: 1) Is there an easier way to approach this ...
0
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2answers
21 views

How do you find a parametric representation for a specific surface?

I trying to find a parametric representation of the plane which goes through the origin and contains the vectors $i-j$ and $j-k$. I found the cross product for these vectors and found that the formula ...
2
votes
1answer
47 views

How to use Cauchy-Scharwz inequality to prove differentiable?

I'm attempting to understand how to prove the function f such that $$f(x,y)=\frac{x^3y}{x^4+y^2}\;if\;(x,y)\neq (0,0)$$ $$f(x,y)=(0,0)\;if\;(x,y)=(0,0)$$ is continuous in $\mathbb R^2$. The solution ...
0
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2answers
30 views

Bilinear Functions on $R^n$

If B is a bilinear function on $R^n$x $R^m$ with real values show that $DB_{(P,Q)}$(h,k)=B(P,k)+B(h,Q) Not sure where to even start, something where the limit of B(h,k)/||(h,k)|| should go to 0 as ...
2
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3answers
61 views

How does algebra on differential forms work?

Let $\omega$ be a 1-form such that: $\omega = a\,dx + b\,dy$ and $\eta$ is a 0-form. I've seen a lot of times where $\eta \wedge \omega$ is written as $\eta \omega$... which kind of makes sense as ...
0
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1answer
11 views

Finding the time derivative of intensity at the moment a particle passes through a point

I'm unsure as to whether I set up the problem correctly. I'm also unsure about whether I was supposed to plug in t= 0 at the very end, considering the particle is at (1,1) when t= 0. Any help ...
1
vote
1answer
26 views

compute sums of x,y given a condition

Problem: given that $\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=p$, then compute $x+y$ try: i tryed to solve by this way $$\begin{align} ...
2
votes
1answer
18 views

How can the derivative of the Euclidean norm be exhibited without considering partial derivatives?

Let $f: \mathbb{R}^n \to \mathbb{R}$ be given by $f(x) = \|x\|$. I would like to show that $Df(a)h = \frac{a\cdot h}{\|a\|}$ without resorting to using partial derivatives. I considered the ...
0
votes
1answer
44 views

Evaluate $\int_{\Bbb{R}^2} e^{-2x^2 -2 y^2 - 2xy} \, {\rm d}(x,y)$

I need to evaluate a double integral with $f(x,y) = e ^{-2x^{2} - 2y^{2} - 2xy }$ with bounds of $x$ and $y$ going from $-\infty$ to $+ \infty$ . I have no idea on how to start. If anybody can help ...
0
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2answers
23 views

Multivariable functions application

We have just started studying functions of several variables and their derivatives and our professor suggested the following problem as food for thought. Two squares, both with length $l=1$ intersect ...
2
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1answer
41 views

How to Evaluate this Multiple integral

$$\int\!\!\!\int\!\!\!\int_{V}\frac{(b-x)\:\mathrm{d}x\:\mathrm{d}y\:\mathrm{d}z}{\left(\sqrt{(b-x)^2+y^2+z^2}\right)^3},(b>a>0)$$ $$V:x^2+y^2+z^2\leq a^2$$ Let ...
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vote
0answers
25 views

Stokes' theorem - how to find dS?

Evaluate $\int_C \textbf{F} \cdot d\textbf{r}$, where $\textbf{F}(x,y,z)= \langle x^2y, \frac{x^3}{3}, xy \rangle$ and C is curve of intersection between the hyperbolic paraboloid $z=y^2-x^2$ and the ...
2
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2answers
38 views

Curl Vector: What exactly is rotating?

I am a little bit confused over the exact conceptual meaning of the curl vector. So I am familiar with the paddle wheel interpretation, but I don't think I am satisfied with that analogy because it ...
1
vote
0answers
30 views

Find Integral $\iint{(x^4-y^4)(e^{xy})dA}$ on D

Where D is in the first quadrant enclosed by $x^2-y^2=3$, $x^2-y^2=4$, $xy=1$, $xy=3$. So my issue is more in the first step. I graphed it and it resembles something I would change to polar ...
1
vote
3answers
72 views

The limit of $e^{\frac{1}{x^2 + y^2}}$ as $(x, y) \to (0, 0)$

$$\lim_{(x, y) \to (0, 0)} e^{\frac{1}{x^2 + y^2}}$$ This really should be a simple limit question, I've done similar things many times before, but I'm very out of practice with limits and cannot for ...
3
votes
2answers
41 views

Two paths that show the limits DNE

I'm having a difficult time trying to find two different paths that give me different limits for the following: $$\lim_{(x,y) \rightarrow (0,0)} \frac{x-y}{x^2 + y^2}$$
1
vote
1answer
34 views

Partial derivatives with $e$

What steps do I need to follow to get the partial derivative with respect to $x$ and separately with respect to $y$ for the following function (I can't seem to get to the correct solution): $$f(x,y) ...
2
votes
1answer
61 views

Showing $\frac{x^2y^3}{x^2+y^4}$ is differentiable everywhere, and if the first order partials are continuous

So the function is defined as above, except at (0,0) the function is defined to be 0. So I computed the first order partial derivatives, and it's obviously continuous at all points other than (0,0), ...
0
votes
1answer
21 views

Surface integral problem $\displaystyle\iint_{S}{x^2dS}$

Let S be the portion of the cylinder $x^2+y^2=4$ between the planes $z=0$ and $z=x+3$. Compute the following integral: $$\displaystyle\iint_{S}{x^2dS}$$ I know the typical conversion formula is: ...
0
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1answer
29 views

Problem with computing numerical Gradient with Matlab

I am playing with numerically computing gradient of some function in matlab, and I have this weird result which I could not figure why. Here is my simple matlab code ...
0
votes
1answer
27 views

Help with this two variable limit.

Determine if the limit exists: $$\lim_{(x,y)\to(0,1)} \frac{1}{x^2+y-1}$$ Based on previous exercises, I'm suppose to prove the inexistence of the limit showing that the limit is different when ...
1
vote
0answers
47 views

Equation of the tangent of the curve

I want to find the equation of the tangent of the curve $f(x,y)=x^5+x^4+y^2$ at $(0,0)$. I have tried the following: $\nabla f (x,y)=(5x^4+4x^3, 2y)$ $\nabla f(0,0)=(0,0)$ The equation of the ...