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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0answers
64 views

Trying to find the tangent plane to a surface

I'm trying to find the tangent plane to the surface defined by $z^2 =x^2 - y^2$ at the point $P(1, 1, 0)$. It seems trivial, but I hit a roadblock because I end up with a line, not a plane! I define ...
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0answers
76 views

negative derivative of integral with time varying domain?

Show that $$\frac{d}{dt}\int_{A(t)}f(x) dx \leq 0 $$ where $f$ is the density function of a zero mean multivariate normal vector with covariance $D=diag(d_1,\ldots,d_p)$ and $A(t)=\{||x+tc||\leq r \ ...
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1answer
74 views

Continuity of a function defined by an integral

Ok, Here's my question: Let $f(x,y)$ be defined and continuous on a $\le x \le b, c \le y\le d$, and $F(x)$ be defined >by the integral $$\int_c^d f(x,y)dy.$$ Prove that $F(x)$ is continuous on ...
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1answer
1k views

Continuity, differentiability and existence of partial derivatives

Here are a few functions whose continuity, differentiability and existence of partial derivatives are to be checked at the origin. I have given the answers, but I would really appreciate it if someone ...
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1answer
61 views

Derivative of integral with time varying domain

Let $f:\mathbb{R}^p \rightarrow \mathbb{R}$ be a smooth function. Let $A(t) \subset \mathbb{R}^p$ be varying with time $t$. Is there a nice expression for $$\frac{d}{dt}\int_{A(t)}f(x) dx$$ ?
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2answers
47 views

Verification for spherical coordinates.

Given the integral to be integrated using spherical coordinates: $$\iiint_{x^2+y^2+z^2 \le z}\sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$$ I used the coordinates as follows: $$ \left\{ \begin{align} x &= ...
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1answer
39 views

How to calculate this mass.

Determinate the mass of portion of spherical surface given by the equation $x^2+y^2+z^2=2a^2$ who is inside of the cylindrical surface given by the equation $(x^2+y^2)=2a^2(x^2-y^2)$, knowing that the ...
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5answers
1k views

Evaluating Double Integral $\int_0^b\int_0^x\sqrt{a^2-x^2-y^2}dy\,dx$

What is the best method for evaluating the following double integral? $$ \int_{0}^{b}\int_{0}^{x}\,\sqrt{\,a^{2} - x^{2} - y^{2}\,}\,\,{\rm d}y\,{\rm d}x\,, \qquad a > \sqrt{\,2\,}\,\,b $$ Is ...
1
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1answer
68 views

Evaluating the surface integral $\iint_\Sigma \mathbf{f} \cdot d \mathbf{a}$ where $\mathbf{f}(x,y,z)=(x^2,xy,z)$

Evaluate the surface integral $\iint_\Sigma \mathbf{f} \cdot d \mathbf{a}$ where $\mathbf{f}(x,y,z)=(x^2,xy,z)$ and $\Sigma$ is the part of the plane $6x+3y+2z=6$ with $x,y,z\geq 0$. I changed ...
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1answer
26 views

Show that the tangent plan pass through the origin

Show that all the tangent plans to the conic surface $z = xf(\frac{y}{x})$ at the point $M(x_o,y_o,z_o)$, where $x_o \neq 0$, pass through the origin of the cordinates First, I've found the tangent ...
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2answers
45 views

Line integral about a circle

Here is the question: Evaluate$$\int Pdx+Qdy$$ where $$P(x,y)=\frac{y+x}{x^2+y^2}$$ and $$Q(x,y)=\frac{y-x}{x^2+y^2}$$ about the circle $$C: x^2+y^2=a$$ oriented clockwise. I tried finding $P_y$ and ...
3
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1answer
67 views

Line integrals of given curves

This question has an integral $$\int(x^4+4xy^3)dx+(6x^2y^2-5y^4)dy$$to be evaluated on the parametric curve $$C:(-(t+2)\cos(\pi t^2), t-1)$$I took the partial derivatives of the terms in the bracket ...
3
votes
1answer
114 views

Triple integral using spherical coordinates

The following function is given: $$\iiint_{x^2+y^2+z^2\leq z} \sqrt{x^2+y^2+z^2}dx\,dy\,dz$$ And I have to calculate this integral using spherical coordinates. The substitutions are standard, I think, ...
3
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3answers
280 views

Tangent plane and Parametrized Surface

"Given a sphere of radius $2$ centered at the origin, find the equation for the plane that is tangent to it at the point $(1,1,\sqrt[]{2})$ by considering the sphere as a surface parametrzed by ...
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1answer
72 views

Multivariate Probability Distribution Expected Value and MGF

Let $X_1, X_2$, and $X_3$ be random variables - continuous or discrete. The joint moment generating function of the random variables is given by \begin{align*} m(t_1,t_2,t_3) = E(e^{(t_1X_1 + t_2X_2 ...
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1answer
163 views

Lagrange Multipliers, two constraints.

Question: . Use Lagrange multipliers to find the constrained critical points of f subject to the given constraints. Here is the equation and the here is my solution. I am stuck now and I don't know ...
1
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0answers
45 views

Constructing a non-degenerate vector-field from old one

This is an attempt to understand Milnor's proof that it's always possible to compute a vector field's index using a non-degenerate field. The strategy is simple: take $z$ to be an isolated zero of the ...
1
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1answer
55 views

Morse index and local minimum/maximum

I have a function in 3-D. The morse index for one of its critical points is 2 and the other is 3. Which one is the local maximum and which is the saddle point?
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0answers
108 views

Maximize the volume obtained by rotating a triangle around one side

Find the triangle with perimeter $2p$ given that, when we rotate it around one of its sides, the solid obtained have the maximum volume. Well, suppose that we have a triangle of sides $a,b,c$ such ...
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1answer
38 views

Show that $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$

Elias M. Stein said that by an application of Green's theorem the following equality holds $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$ where $\Delta _{S}$ is a ...
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0answers
38 views

Equation for tangent plane

The function is as follows: $$\begin{bmatrix}u \\ v \end{bmatrix}= f(x,y,z)= \begin{bmatrix}e^{x-y}+y \\ \sin(x^2-z) \end{bmatrix}$$ and I need to obtain the equation of the tangent plane. The ...
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1answer
72 views

Morse index and degeneracy

The function is as follows: $$f(x,y,z) = e^x(xy-y^2-z^2)$$ I have found the critical points to be $(0,0,0)$ and $(-2,1,0)$. The question asks to determine the morse index of the points and the ...
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2answers
688 views

Projection of ellipsoid

Find the projections of the ellipsoid $$ x^2 + y^2 + z^2 -xy -1 = 0$$ on the cordinates plan I have no idea how to do this. I couldn't find much on google to help me with it too. Thanks in ...
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1answer
144 views

Prove that there exists a scalar potential $f( \mathbf{ x} )$ such that $\mathbf{ F } = − \nabla f$ [2012 11c]

Question: If F is an irrotational vector field (i.e. $ \nabla \times \mathbf{ F = 0 }$ everywhere), prove that there exists a scalar potential $f( \mathbf{ x} )$ such that $\mathbf{ F } = − ...
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1answer
30 views

Partial derivatives of $f(x_1, \dots, x_n, g(x_1, \dots, x_n))$

I have some arbitrary differentiable function of $n+1$ variables, $f(x_1, \dots, x_n, y)$, and some other arbitrary differentiable function of $n$ variables, $g(x_1, \dots, x_n)$. I then define a new ...
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2answers
60 views

Lagrange Multipliers, optimization question.

Here is the question. And here is my attempt at the solution: $ h(x, y, z, \lambda) = 4x^2 + yz + 15 - \lambda(x^2 + y^2 + z^2 - 1) = 0 $ $$\\ $$ The partial derivatives of h: $$ \\ h_x = 8x - ...
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0answers
99 views

Expectation of $n$-dimensional Inverse Bessel Process

I think the main problem for me is to calculate the integral of $$\int_{0}^{\infty}\frac{e^{-\frac{r^2}{2t}}}{\sqrt{x^2+r^2}}r^{n-1}dr,n\geq2$$ For n=2, change of variable $y=\sqrt{x^2+r^2}$ would ...
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0answers
130 views

Partial Derivative of a Scalar-product Resulting from Vector Multiplication

I am trying to differentiate the function below, but I am running into problems due to the point-wise multiplication with the matrix. $f(x,y,A) = (x^{T}y) \cdot A$ Where $\cdot$ denotes point-wise ...
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2answers
51 views

Prove that $f(x+h)-f(x) - \langle\nabla f(x), h\rangle\geq 0 \Rightarrow f $ convex

At this link there is a demonstration that for $f$ continuously differentiable on $C \subseteq \mathbb{R}^n$ convex, $f(x+h)-f(x) - \langle\nabla f(x), h\rangle\geq 0 \Rightarrow f $ convex. This ...
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3answers
427 views

Show discontinuity of $\frac{xy}{x^2+y^2}$

How to show this function's discontinuity? $ f(n) = \left\{ \begin{array}{l l} \frac{xy}{x^2+y^2} & \quad , \quad(x,y)\neq(0,0)\\ 0 & \quad , \quad(x,y)=(0,0) \end{array} ...
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3answers
146 views

show continuity of $\frac{xy}{\sqrt{x^2+y^2}}$

How to show this function's continuity? My book's 'Hint' says $|xy| \leq \frac12(x^2+y^2)$ can be used. $ f(x,y) = \left\{ \begin{array}{l l} \frac{xy}{\sqrt{x^2+y^2}} & \quad , ...
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1answer
298 views

Parameterization of curves for line integrals

The question asks to find the following line integral $$\int |y|ds$$ along the curve $$C: (x^2+y^2)^2=8^2(x^2-y^2)$$ I used the parameterization $$x=\sec m; y=\tan m$$ and solved it further to get ...
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2answers
67 views

Minimum and maximum values

I am not sure if my method for this question is correct: Given the function $$f(x,y)=x^2+y^2+2x+y$$, find it's minimum and maximum values about a closed disc of radius 2 centred at the origin. I ...
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2answers
31 views

how to find the limit of this function?

$f(x,y)= \frac{( 1+ x^2 y^2)^{1/3} - 1}{x^2 + y^2}$ for $(x,y) \neq (0,0)$ 〖Lim〗_((x,y)→(0,0) ) f(x,y) I have tried many paths but the limit is becoming $1/0$. so what is the solution to this ...
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1answer
22 views

Multivariable calculus: determining $dV$ when $0 \leq ax+by+cz \leq l$, $0 \leq dx+ey+fz \leq m$ and $0 \leq gx+hy+iz \leq n$

As my previous question had severe typos, I deleted my question and re-asking. Suppose that we are solving a multivariable calculus problem. The region that we are integrating over is described by $0 ...
4
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1answer
28 views

Show that $\sum_{cyc} J(x,J(y,z))=0$.

Let $x,y,z$ be functions of $(u,v)$ and $J$ be the Jacobian matrix. Show that $\sum_{cyc} J(x,J(y,z))=0$. I expanded the thing and realized that the first term in the sum is ...
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1answer
59 views

How does a fixed point of a dynamical system change when the system is changed slightly?

I've come across this problem in the course of my work, and I'm a bit stuck on it. Suppose I have an $n$-dimensional smooth dynamical system $$ \dot{x_i} = f_i(\mathbf{x}), $$ and suppose there is a ...
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2answers
35 views

$\iint_V |y-x^{2}| \operatorname{d}x \operatorname{d}y$ with $V = [-1,1] \times [0,2]$

it's especially difficult because i don't understand how to integrate absolute value terms. I only know that if you function, say $x^{2}-1$, is below the $x$-axis i need to integrate $1-x^2$ between ...
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1answer
31 views

Partial derivative techniques

The question is as follows: $$ r = y^2, s = xy, t = -x^2, u = f(r,s,t)$$ , such that $f$ has continuous second order partial derivatives. Represent $u_{xy}$ in the following format.$$ u_{xy} = ...
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1answer
47 views

How do I find the highest and lowest points made by the union of these two functions using Lagrange Multipliers?

Find the highest and lowest points made by the union of these two functions using Lagrange Multipliers. $x^2+y^2+z^2 = 16$ $(x+1)^2+(y+1)^2+(z+1)^2 = 27$ I got the basics down, I used the first ...
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2answers
80 views

Is a continuously differentiable function convex if all its partial second derivatives are non-negative?

I'm having trouble understanding the relevant Wikipedia article which begins with a convex set $X$ and then uses functions of single variables for succeeding examples; the MathWorld article seems to ...
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1answer
56 views

If $T$ is injective then there exists $\alpha>0$ such that $||Tx||\geq \alpha||x||$

Is this proof correct? I'm proving that if $T$ is a linear operator whose is injective then exist $\alpha>0$ such that $$||Tx||\geq\alpha||x||$$ for all $x$. By contrapositive. Assume that for all ...
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1answer
53 views

Is that proof correct? I'm proving that the set of injective Linear transformations in open.

Let $T$ an inyective linear transfomation from $\mathbb{R}^n$ to $\mathbb{R}^m$. Then there exist an $\alpha>0$ such that $$||T(x)||\geq \alpha||x||$$ for all $x$. Let $S$ a linear transformation ...
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2answers
234 views

Why is every continuously differentiable function with a uniform bounded derivative lipschitz continuous

I only know how to prove this for functions on a convex set by using the mean value theorem, but is this also true for this general case when nothing is said about the domain of the function besides ...
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1answer
42 views

Steps to find gradient of this?

I don't understand how to get from .54 to .55, any help? Edit: Wolfram alpha gives the answer as this http://www.wolframalpha.com/input/?i=grad+Ucos%28theta%29%28r%2Ba%5E2%2Fr%29 I can't seem to ...
0
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0answers
69 views

Prove that the set of injective linear transformations is an open set.

Prove that the set of injective linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^m$ is an open set. Using the fact that a Linear transformation is injective if and only if there is ...
2
votes
5answers
163 views

Implicit differentiation

I want to differentiate $x^2 + y^2=1$ with respect to $x$. The answer is $2x +2yy' = 0$. Can some explain what is implicit differentiation and from where did $y'$ appear ? I can understand that ...
1
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1answer
256 views

using total derivative to approximate changes

I have the function $ f(x,y,z) = xyz$ with $y = x^2 $, $z = \sqrt[3]{x} $ The initial values are $ (27, 729, 3) $ I want to compute the total derivative of f wrt x and use that to approximate the ...
1
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1answer
65 views

Polygon Collision Detection Function using Dirac Delta Distribution and Divergence Therom, help.

I would like to find a way of doing polygon collision detection that handles concave/irregular polygons in a very performant matter. I would be great if you guys could check my math and reasoning up ...
1
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1answer
128 views

Minimum area of the parallelepiped surface

Among all the retangular parallelpipeds of volume $V$, find one whose total surface área is minimum Using the Lagrange Multipliers method, I've found that it is a cube with dimensions $ \sqrt[3]{V} ...