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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1
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1answer
147 views

Double Integral $\iint_D\ (x+2y)\ dxdy$

$$\iint_D (x+2y)\ dxdy $$ If the area is range by $x=2,\ x=3,\ y=x,\ y=2x$, how to include the lines? How limits for integral will looks like? You mean something like this? ( I made mess) $$\iint_D ...
4
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2answers
111 views

Double integral: $\int_{y=0}^1\int_{x=y}^1 e^{\large x^2}\ dx\ dy$

Could someone help me with this question? I am stuck on it. Compute the following double integral: $$\int_{y=0}^1\int_{x=y}^1 e^{\large x^2}\ dx\ dy.$$ How to compute the integral when the inner ...
2
votes
1answer
139 views

Integration w/ Change of Variables

folks. I've got this question: Let $D$ be the region $\{(x,y) ~|~ 0 \leq y \leq x, 0 \leq x \leq 1\}$. Evaluate: $$\iint_D (x + y) dxdy$$ by making the change of variables $x = u + v$, $y = u ...
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2answers
117 views

Help my homework: $\int_0^1\int_{0}^1\frac{x^2-y^2}{(x^2+y^2)^2}dy\, dx$ [duplicate]

I am trying to integrate $$\int_0^1\int_{0}^1\frac{x^2-y^2}{(x^2+y^2)^2}dy\, dx$$ In my book said that use tangent function but I don`t know how to evaluate it. Please help me. I want to know the ...
3
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2answers
135 views

Computing the inverse Fourier transform of $\frac{1}{1+|\xi|^2}$ for $\xi \in \mathbb{R}^n$.

I'm trying to compute the integral $$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$ I know that for an integral like $$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + ...
0
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2answers
82 views

Boundaries for area with double integration.

I have to find the area using double integral for the domain bounded by $$y^2=x$$ and $$y=x-2$$ Now, I want to find my integral boundaries: I did $y=x^2, y=x+2$, solved this system and get ...
1
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0answers
18 views

Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is: ...
0
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1answer
49 views

Implicit function derivation

I have function $h(x,y)=e^{xy^2-1}+\log{\frac{x}{y}}-1$ and I have to find if a function $y=f(x)$ around $[1,1]$ exists. I have to check some conditions in order to find out if $y=f(x)$, ...
1
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0answers
40 views

Intuition for Directional Derivative Equation

Given some function $ z=f(x,y) $, the directional derivative can be used to calculate the rate of change of $ z $ in the direction of some unit vector $ \vec{u} = <a,b> $. The directional ...
2
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1answer
59 views

Reference Request: How to Parametrize Curves and Surfaces in $\Bbb R^3$

I don't feel like I have a good grasp of how to parametrize a curve or surface. I can quickly enough verify that a given parametrization DOES correspond to a curve, and I've memorized a few of the ...
0
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2answers
23 views

Determining when the gradient of a function is parallel to a vector.

Let $G(x,y,z) = \left( \sqrt{x^{2} + y^{2}} - R \right)^{2} + z^{2}$. If my calculations are correct, then $$\nabla{G} = \left(x \left(2 - \frac{2R}{\sqrt{x^{2} + y^{2}}} \right), y \left(2 - ...
1
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3answers
55 views

Find the time at which the particle has traveled $14$ units

If a particle follows the path defined by $$r(t) = (2t^{3/2},2t+1, \sqrt{5} t )$$ and starts at $t=0$, at what time will the particle have traveled a total of $14$ units ?
0
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1answer
18 views

Find the tangent line

Find the tangent line to the curve $\displaystyle r(t)= \left(3\ln t,2t-3,\frac1t \right)$ at $t=1.$ I know that I have to find the derivative of $r(t)$ but I do not know the following steps.
0
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2answers
32 views

Find the parametric equation

Find the parametric equations from the following curves A. The line segment from $P=(9,8,5)$ to $Q=(13,-2,0)$ B. $x^2+y^2=9$ for only positive $x$ values. I could not even get to the starting point ...
0
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1answer
35 views

Double integral area.

I have to do $$\int\int_D{\rm d}x\,{\rm d}y$$ where D is between $y=2x^2-2$ and $y=1-x^2$. I draw my area and my new integral become: $$\int_{1-x^2}^{2x^2-2}\int_{-1}^{1}{\rm d}x\,{\rm d}y$$. I ...
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0answers
25 views

What's the jacobian of this function

We are given a function $f$ that maps the coefficients of a polynomial to its roots. meaning $f(a_1,a_2,a_3)=(x_1,x_2,x_3)$ if ...
0
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1answer
24 views

How do i evaluate $\int_{|(x,y)|=r} \frac{xy^2}{x^2+y^2} dy$?

How do i evaluate $\int_{|(x,y)|=r} \frac{xy^2}{x^2+y^2} dy$ ? By Green's theorem, i have shown that it is equal to $\frac{1}{r^2}\int \int_{B(0,r)} y^2 dxdy$ I don't know what should come next..
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1answer
50 views

Understanding surface integrals

My question is a bit vague, but I'm trying to get a better understanding of surface integrals and their relation to physics. Suppose I have a surface, say a sphere, and I have a function which gives ...
0
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1answer
59 views

Problems in Partial Derivatives

I'm trying to compute the Partial Derivatives for the equation: $$k(a, b) = a^2 \exp\left(-\frac{1}{2b^2} (x-y)^2\right)$$ w.r.t. $b$ $x$ and $y$ are known; both $a$ and $b$ are hyper-parameters. ...
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0answers
38 views

Proving a vector field is conservative

Let $\vec{F}=(P,Q) \in C^1 (\mathbb{R}^2 - (0,0) ) $ such that $Q_x=P_y $ and let $\gamma$ be a closed line surrounding the origin such that $\int_{\gamma} \vec{F} \cdot \vec{dr} =0 $ . Show that ...
0
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3answers
62 views

Why and when $\lim_{r\to0}\int_{\partial B(x,r)}u(y)\;dS(y)=u(x)$?

Let $U\subset\mathbb{R}^n$ be an open set, $x\in U$ and $u\in C^2(U)$ a harmonic function. I would like know what is the theorem that is used to conclude that $$\lim_{r\to0}\int_{\partial ...
3
votes
2answers
97 views

What does $\iint_S (\hat{\mathbf{n}} \times \nabla \psi) \; dS$ mean?

In the Wikipedia article on vector calculus identities, we have the following $$\oint_{\partial S} \psi \; d\mathbf{\ell} = \iint_S (\hat{\mathbf{n}} \times \nabla \psi) \; dS$$ The right hand side ...
1
vote
1answer
36 views

The Laplacian and a nice PDE

Given the Laplacian: $$\Delta u= \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} $$ I had to show that by using this $$v(r,\theta ):=u(r\cos \theta ,r\sin \theta ) $$ I can ...
0
votes
2answers
459 views

Distance from Ellipsoid to Plane - Lagrange Multiplier

Find the distance from the ellipsoid $x^2 + y^2 + 4z^2 = 4$ to the plane $x + y + z = 6$. I'm trying to do it using Lagrange multipliers over the distance equation, but then it just gets ...
0
votes
1answer
269 views

Finding the average value of a function! over a region!

Completely related to this question: Finding surface area of part of a plane that lies inside a cylinder??? Find the average value of the function $f(x, y, z) = x^2yz$ over $S$ How does one do ...
2
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2answers
40 views

About the order of integration for double integrals

I have to compute $\int\int (2x-y) \,dx \, dy $ on the domain $\{ (x,y) \in R^2 : 1\leq x\leq 4, 0\leq y\leq \sqrt{x} \}$ So mi first try is to do: $\int_0^{\sqrt{x}}\int_{1}^{4} (2x-y)\, dx \, dy ...
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2answers
32 views

multiplying 2 vectors using cross product

so i'm trying to get (-1,2,-1) and (1,1,-2) multiplication into a new vector book says (5,-3,1) unfortunately it showed us how to do 2x2 and 3x3 matrixes and I learned how to apply those ...
1
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0answers
29 views

Is there a diffeomorphism $f$ such that $V(B) < V(f(B)) <V(B)+V(B)^2$

Simple to understand calculus question that involved change of variable theorem in integration. Suppose $B$ is some open ball in $\mathbb R^n$, and $f: \mathbb R^n \to \mathbb R^n$ is a ...
0
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1answer
27 views

Center of Mass double Integral using polar Coord.

Find center of mass given Lamina pictured: https://s3.amazonaws.com/wamapdata/qimages/qtrring.gif with inner radius of 3 and an outer radius of 7, and a density function $$\rho(x,y) = ...
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0answers
46 views

Stoke's theorem explanation

Firstly, Wikipedia informs me that Stoke's theorem might mean two different things so I'm pretty sure I'm talking about the general one. I'll just state what I understand from it and I'd like someone ...
0
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0answers
35 views

Taylor expansions, inequalities and more

(Part A) I have to find the Taylor expansion of order 2 around (0,0) of $$f: \mathbb{R}^{2}\rightarrow \mathbb{R}$$ $$(x,y)x \mapsto f(x,y) = x\log (1+y)+sin(x+y) $$ Furthermore I have to prove if ...
1
vote
1answer
27 views

Solve system to find critical points.

Hi I have to find the stationary points for $$f(x)= x^4+y^4-(x-y)^2.$$ So far i founded the partial derivatives for $x$ and $y$. Next step is to solve this system to get my critical points: $$ ...
0
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2answers
3k views
0
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1answer
42 views

Prove that $\oint_{\partial S} \psi \; d\ell = \iint_S (\hat{\mathbf{n}} \times \nabla \psi) \; dS$

In the Wikipedia article on vector calculus identities, we have the following $$\oint_{\partial S} \psi \; d\ell = \iint_S (\hat{\mathbf{n}} \times \nabla \psi) \; dS$$ How do I prove this? I tried ...
2
votes
1answer
26 views

Acquiring $Df(\mathbf{x})$

Sorry for the probably easy and silly question, but I try to teach myself linear algebra and I am stucked at "the derivative as a matrix" part. I know how to differentiate partially and I know how ...
0
votes
1answer
43 views

Finding the critical points of a function

What are the steps in finding the critical points of a function in general? Say for example, the function $$f(x, y) = 2x^3 + 11x^2 + 0.5y^2 - 2xy$$ I can't quite seem to understand the steps/method ...
0
votes
1answer
72 views

Finding max/min through lagrangian

I am trying to solve this problem, but I am doing something wrong: $$f(x,y,z)=x^2-y^2,M=\{[x,y,z]\in\mathbb{R}^3:x^2+y^2+z^2=9,x+z\ge1\}$$ And let $g(x,y,z)=x^2+y^2+z^2-9$. Set M is closed and ...
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vote
2answers
68 views

Surface: intersection of 2 polar curves

I have these two polar curves: $$ C_1: r = 2 - \cos(\theta)\\ C_2: r = 3 \cos(\theta) $$ Plots: C1 and C2. I need to find the surface of $D = C_1 \cap C_2$. I started by finding the solution to ...
0
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1answer
2k views

Slope of the indifference curve

So I am taking mathematical economics and in the HW my professor asked to draw a couple of level curves for $f(x,y)=xy$. Attempt: So I did it the following way. To find the slope I took the ...
0
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2answers
124 views

Can a saddle point have a horizontal tangent plane?

I have saddle points, where the partial derivatives both equal zero, but intuitively it's hard to understand how they could only touch one point on the surface. Please help me brain? Are there ...
7
votes
4answers
206 views

Calculate $\int_{0}^{1}\frac{\arctan(x)}{x\sqrt{1-x^2}}dx$

I am preparing for a calculus exam and I was asked to calculate $$\int_{0}^{1}\frac{\arctan(x)}{x\sqrt{1-x^2}}dx$$ Using the hint that $$\frac{\arctan(x)}{x}=\int_{0}^{1}\frac{dy}{1+x^2y^2}$$ I ran ...
2
votes
2answers
57 views

Epsilon delta prove for continuïty$ (1-\cos(|xy|))/y^2$

Let a function, $\mathbb{R}^2\to\mathbb{R}: \begin{Bmatrix} \frac{1-\cos(|xy|)}{y^2}&y\neq0\\ \frac{x^2}{2}&y=0 \end{Bmatrix} $ I have to prove this is continious. For y$\neq 0$, this is ...
0
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3answers
44 views

Find the maximum value of the following function?

$ f(x,y) = x^2 + y^2 + xy + \frac1x + \frac1y $ Find the extremum values of the above function. For your reference, answer to the problem is - minima at $(3^{-1/3},3^{-1/3})$ But how? Please help! ...
2
votes
1answer
68 views

Derivative of $f(a)=\int_{0}^{1}\sin(t\cos(a)) \log(t)dt$

Fairly simple question regarding integral dependant on parameter. we have $f(a)=\int_{0}^{1}\sin(t\cos(a)) \log(t)dt$ We want to find $f'(a)$. I tried to do this using Leibniz Rule: ...
2
votes
0answers
43 views

Difficult integrals, do they converge, show there's no dependence on parameters.

I am trying to figure out whether these integrals: a) $$\int_{\mathbb R^2}{{\rm d}\xi \over \left\vert\vphantom{\Large A}\,\log\left(\left\vert\,x - \xi\,\right\vert\right) -\log\left(\left\vert\,y ...
0
votes
1answer
58 views

Does the rank of the Jacobian contain information about invertibility.

Let $m\geq n$ and consider $f: \mathbb{R}^n \to \mathbb{R}^m: x\to f(x)$ continuous and differentiable. Does the rank of the Jacobian of $f$ (at a point $x_0$) contain any information of whether it ...
1
vote
1answer
48 views

Integral notation from cartesian from polar coordinates

Given an integral $$I=\int\limits_{\mathbb{R}^n} \cdot \; dx,$$ we can introduce polar coordinates, such that $$I=\int\limits_{\Bbb S^{n-1}} \cdot \; d\theta.$$ Another way to express the latter one ...
0
votes
3answers
44 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
1
vote
1answer
179 views

Extreme value Lagrange multiplier (max or min?)

I am to determine the the range of the volume of a tetrahedron enclosed by the coordinate axes and a tangentplane on the ellipsoid $x^2 + 2y^2 + 3z^2 = 1$. The volume of the tetrahedron can be derived ...
1
vote
1answer
29 views

Double integral boundaries.

I have to find the area using double integral for the domain bounded by $$y=x^2$$ and $$x-y+2=0.$$ Now, I want to find my integral boundaries: I did $y=x^2, y=x+2$, solved this system and get ...