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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1answer
21 views

proving one definition of a conservative field (need help FTC for multivariable case)

If f(x,y,z)= $\int_{c} (F.dl)$ ,to prove that grad f= F is used the path from the origin to a generic point (x,y,z). And first we move along the x axis then y axis then z axis.therefore f(x,y,z)= ...
2
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0answers
29 views

$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$ is differentiable?

Let $ D=\{(x,y) \in \mathbb{R}^2 : x>0, y>0\}$. Show that the function $$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$$ is well defined on $D$. ...
0
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1answer
23 views

Global maxima and minima of $f(x,y) = xy$

I wanted to check with you if my reasoning to this problem was right. Find the global maxima and minima of $f(x,y) = xy$ inside the set $A = \{ (x,y) \in \mathbb{R}^2: \frac{|xy|}{|xy|+1} \leq ...
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3answers
18 views

Global extremes of $f(x,y) = \sin(xy)$ in $A$

I am trying to solve the following problem: Find the extremes of $f(x,y) = \sin(xy)$ restricted to points of $A = \{ (x,y) \in \mathbb{R}^2: x^2+y^2 = 1\}$. My first attempt was to use Lagrange ...
0
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1answer
26 views

Double Integral requiring polar coordinates

I've tried my best in solving the following double integral. I just wanted to be sure I've done the right thing here (b is the vertical axis whilst a is the horizontal axis). ...
1
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1answer
31 views

Global max./min. of $f(x,y) = y+x-2xy$

I am having some doubts about this exercise: Find maximum and minimum of $f(x,y) = y+x-2xy$ restricted to the interior and border of $R = \{(x,y) \in \mathbb{R}^2 : |x| \geq \frac{1}{2}\}$. I ...
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0answers
12 views

Approximation of sets by sets with regular border

What kind of conditions on a (bounded) set $E \subset \mathbb{R}^{n}$ ensure that it can be approximated from outside/inside by sets with regular border (say Lipshitz or $C^{k}$ conditions) in the ...
1
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2answers
28 views

Find $f(x,y)$ if each level curve $f(x,y)=C$ is a circle centered at the origin and having radius

Find $f(x,y)$ if each level curve $f(x,y)=C$ is a circle centered at the origin and having radius (a) $C$ (b) $C^2$ (c) $\sqrt{C}$ (d) $\ln C$. I am not good at this...
0
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0answers
40 views

Plane tangent to an ellipsoid

I'm having bit of a difficulty with this problem. I have answer, there was also text how they solved it, I've seen other ways to solve this but I never get the desired answer. Perhaps you can point me ...
1
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0answers
17 views

Verification of stkokes theorem

Verify Stokes’ Theorem for $$v = (y − z + 2)\vec{i} + (yz + 4)\vec{j} − xz\vec{k}$$ where $S$ is the surface of the cube bounded by $x = 0$, $y = 0$, $z = 0$, $x = 2$, $y = 2$, $z = 2$ with the face ...
0
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0answers
47 views

Infinite stationary points?

If we have a function involving $\cos x$ and if this is the only part where $x$ exists, so for example something like $$y\cos x$$ og $$y(\sqrt{y}+\cos x)$$ does that mean we'll always have an infinite ...
1
vote
1answer
75 views

nice name for the image of multivariable function

Consider a differentiable function $f:D\subset\mathbb R^m\mapsto \mathbb R^n$ with $m\le n$. I know if $m=1$ then $f(D)$ is called by "path", if $m=2$ then $f(D)$ is called by "surface" and if $m=3$ ...
0
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1answer
283 views

In what order should I learn linear algebra and multivariable calculus?

I took AP calculus in high school and I really enjoyed it, but when I got to my university I was upset that I couldn't take Calculus II as it didn't fit in my schedule. I feel kind of behind now that ...
0
votes
1answer
158 views

Related Rates Cylinder

a) Assuming even distribution of oil, calculate the volume in cubic meter oil slick when the radius is 1 km and the height is .23 meters B) AT the exact instant in part a, the radius is increasing at ...
1
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0answers
62 views

Gradient of the function and the contour line

I do not understand, reading the chapter in the book about Lagrange multipliers, why the gradient of the function $f$ is perpendicular to the contour line? There is no sufficient explanation there, ...
2
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2answers
57 views

Evaluate a definite double integral of the integrand of the gaussian integral

I have to solve this by changing the order of integration, $$\int_0^4 \int_\frac y4^2 e^{-x^2} \,dx\, dy$$ and I got this far, $$\int_0^1 \int_0^{4x} e^{-x^2}\, dy\, dx$$ but I'm pretty sure I'm ...
3
votes
1answer
41 views

Show that the function f solves the homogenous wave equation

This is a cleaned up and refined repost of my previous attempt, after I did some research on the subject. SOLVED The only problem here was that I was making tons of little mistakes, always watch ...
0
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0answers
94 views

Determine the set of points at which the function is continuous:

Determine the set of points at which the function is continuous: $f(x,y,z) = \sqrt{y-x^2} \cdot \ln z$. I know that $y-x^2$ must be $\geq 0$ (because of the square root). I also know that $z>0$ ...
1
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1answer
102 views

Beginner exercise on the Implicit function theorem

My first exercise on the Implicit function theorem: Show that the non-linear equation $x^4 + e^y + sin(z) + z^5 = 1$ has a "local resolution function" $(x,y) \rightarrow z(x,y)$ (is this the right ...
1
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1answer
73 views

Show that f solves the so called wave equation

Task $\text{Let } \; c \in \mathbb{R} \; \text{ be a given parameter, with } \; c > 0$ $\text{ Show that } \; f: (\mathbb{R}^3 \setminus \{ \vec{0} \}) \times \mathbb{R} \to \mathbb{R} \; ...
0
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1answer
49 views

What is the sub-differential of the separable sum $R(w) = \sum^{d}_{j=1} |w_j|$?

Recall the definition of a sub-differential: $$\partial F(w_0) = \{ v : \forall w, F(w)-F(w_0) \geq v \cdot (w - w_0)\} $$ Intuitively, for any w in the domain of the function one can draw a plane ...
1
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1answer
27 views

Line integrals of a vector field

I had no problem with part (a) or (d). The sketch was fine, and if we were given the F in part (d), we can use Stokes' Theorem. However, I am struggling with (b) and (c). For (b) I know I must ...
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0answers
30 views

Is there a good textbook/book out there that explains sub gradients thoroughly?

I was interested in learning and understanding sub gradients as much as I could from some good resource. I know what the definition is, but I seem unable to apply the definition to prove basic facts ...
10
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2answers
162 views

Closed form of $\displaystyle\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}}\, dy\, dx$

Can the following double integral be evaluated analytically \begin{equation} I=\int_{0}^{\Large\frac{\pi}{4}}\int_{\Large\frac{\pi}{2}}^{\large\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin ...
0
votes
1answer
33 views

Surface Area Using Double Integrals

I need to find the surface area of the function: $$z = y^2-x^2$$ Between the cylinders: $$x^2+y^2=1$$ $$x^2+y^2=4$$ So far I've tried converting it into polar coordinates to no avail and I don't ...
1
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0answers
96 views

Evaluate the triple integral bounded by a cylinder and planes

Evaluate the triple integral $\displaystyle\iiint\limits_{W}z\,dx\,dy\,dz$ over the region $W$ which is bounded by the planes $x=0$, $y=0$, $z=0$, $z=1$, and the cylinder $x^2+y^2=1$ in the first ...
4
votes
1answer
68 views

Find K values that make a differential equation solution stable

Given some differential equations, ie. "a", or "b": a. $$Y'''+Y''+2Y'+KY=0$$ b. $$Y'''+KY''+3KY'+2Y=0$$ How do I get the $K$ values that make the solution stable? I know that for "a", it should be ...
0
votes
1answer
53 views

Evaluate the triple integral of $x^2+y^2$ where D is a pyramid

Evaluate the triple integral of $f(x,y) = x^2+y^2$ where the region of integration is the pyramid with top vertex at $(0,0,1)$ and base vertices at $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, and $(1,1,0)$. I ...
1
vote
1answer
63 views

Centroid of a solid semiellipsoid

Assuming the result that the centroid of a solid hemisphere lies on the axis of symmetry 3/8 of the way from the base toward the top, show, by transforming the appropiate integrals, that the center of ...
1
vote
2answers
166 views

Find volume above cone within sphere

My objective: Using spherical coordinates, set up and compute an integral to find the volume of the ice-cream-cone shaped solid lying above the cone $z = \sqrt{x^2 + y^2}$ and below the sphere ...
5
votes
1answer
273 views

Find center of mass of triangle with varying density

My objective: Find the center of mass of a thin triangular plate bounded by the y-axis and the lines $y= 7x+3$ and $y= 36-4x$. Assume that the density is given by $\delta(x,y) = 7x+2y+2$. In ...
0
votes
2answers
77 views

Show that the function satisfies the given partial differential equation.

Show that the function satisfies the given partial differential equation. The function is $z=\sqrt{x^2+y^2}$ and they want me to show $x \cdot \displaystyle\frac{\partial z}{\partial x}+y \cdot ...
0
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1answer
2k views

Solve the following differential equations by converting to Clairaut's form through suitable substitutions.

The question comprises of three subparts which need to be converted to Clairaut's form through suitable substitutions and then solved : (a) x p2 - 2yp + x + 2y = 0 (b) x2 p2 + yp (2x + y) + y2 = 0 ...
1
vote
1answer
31 views

How to compute the integral of that area?

The area is given by $ 0 \le x+y \le 4-(x-y)^2 $ ? By magic the inequalities were transformed into $ 0 \le \sqrt 2 \ u \le 4 - 2v^2 $ and after that computing the integral became almost trivial. I ...
0
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0answers
110 views

Best approximation of arc length of curve with only two line segments

I know that arc length can be approximated with a certain number of line segments, as shown in this picture. http://en.wikipedia.org/wiki/File:Arclength.svg The sum of the lengths of the line ...
2
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0answers
46 views

Compute the Jacobian of the following

The question is this: Consider the parabaloid $\{(x,y,z)|z=1-x^2-y^2\}$, let $A$ be the subset satisfying $z>0$. Consider the plane $\pi$ given by $z=1$. The functions $x$ and $y$ act as ...
0
votes
1answer
192 views

Prove that there are an infinite amount of critical points and they are all local (and absolute) minimums on x^2+4y^2-4xy+2

I am studying for a fast approaching Calc 3 midterm exam and ran into this problem in the textbook. Show that $f(x,y) = x^2+4y^2-4xy+2$ has an infinite number of critical points and that $D = 0$ ...
1
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1answer
35 views

contour which can be homeomorphic?

If I have a function $\phi:\mathbb{R^{2}}\rightarrow\mathbb{R}$ which is $C^{\infty}$ without critical points, can I assure that all the contour are homeomorphic?
1
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1answer
41 views

Multivariable continuity and partial derivatives question

P1 : Given $f(x) = x$ if $y=0$, $x - (y^3)\sin(1/y)$ when $y$ is not equal to $0$. Is $f$ continous and differentiable at at $(0,0)$? P2. Given $f = x \sqrt{x^2 + y^2} / |xy|$ when $x$ is not ...
2
votes
1answer
58 views

Can $f(x,y) = |x|^y$ be be made continuous?

Can $f(x,y) = |x|^y$ be appropriately defined at (0,0) in order to be continous there . if we approach from path y=mx then f(x) becomes |x|^mx with x approaches to 0 .then by taking logs and we ...
2
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3answers
57 views

Proving that limit exist

To show that limit exist and is equal to zero $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2y + xy^2)}{xy}$$
2
votes
1answer
661 views

Calculus on Manifolds (Spivak), problem 2-41(a)

Let $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ be differentiable. For each $x \in \mathbb{R}$ define $g_x:\mathbb{R}\to\mathbb{R}$ by $g_x(y) = f(x,y)$. Suppose that for each $x$ there is a unique ...
1
vote
1answer
34 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
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0answers
61 views

Analyzing differential equations system stability via phase diagram

I´m having a hard time trying to understand how to analyze stability using the phase diagram method for systems, could you please guide me? My result should be just knowing if we´re in front of a ...
1
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2answers
61 views

Question related to Lagrange multipliers

I am stuck with the following problem: A is symmetric $n\times n$ matrix and $f(x)=(Ax)x$ for $x\in {\bf R}^n$. I need to show that the maximum and the minimum values of $f$ on the unit sphere ${x: ...
2
votes
3answers
242 views

Using Lagrange multipliers to solve for minimum

I am having troubles with one part of this homework problem. Hopefully somebody can help me out: Find the minimum and maximum values of the function $f(x,y)=x^2+y^2$ subject to the given constraint ...
0
votes
1answer
369 views

Splitting Up Integrals and Multiplying Them

$$I_x = \int_0^b\int_0^h\rho y^2\,\mathrm{d}y\mathrm{d}x$$ So here's the current problem I'm working on, just for an example. I saw my teacher break up a triple integral in class today then multiply ...
0
votes
1answer
24 views

Limit of multivariate polynomial with large arguments

If I have a polynomial $f(x,y)=x^4+y^4-4xy$, how would I go about showing that as the standard norm of $(x,y)$ goes to infinity, $f(x,y)$ goes to infinity?
3
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2answers
112 views
4
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2answers
172 views

to find the extreme values of function..

can anyone just help me with the below stated problem: Show that: $1.)$ $\text{sin}(x)+\text{sin}(y)+\text{sin}(x+y)$, $x,y\in [0,\pi/2]$ has a global maximum $3\sqrt3/2$ at ...