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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194 views

Do we need additional assumptions for problem 3-37 (b) in Spivak´s calculus on Manifolds?

Problem 3-37 (b) reads: Let $A_{n}=[1-1/2^{n},1-1/2^{n+1}]$. Suppose that $f:(0,1)\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for any $x\notin$ any $A_{n}$. Show that ...
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1answer
156 views

Level sets of function

Let $f$ and $g$ be functions $\mathbf{R}^3\to\mathbf{R}$. Assume $f$ is differentiable and $f'(\mathbf{x})=g(\mathbf{x})\mathbf{x}$. Show that $f$ is constant on spheres centered at the origin. ...
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0answers
149 views

Partial derivatives using variables after a transformation

I have a transformation $$(x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2))$$ and I wish to find $$\partial x'_1\over \partial x'_2$$ how might I evaluate this? If it is difficult to find a general expression for ...
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0answers
114 views

How simple is it to solve this Differential Equation

How to solve this Differential Equation? How simple is it to solve this Differential Equation? Any guidelines? Any hint? How to approach the solution? Have anybody seen things like it before? ... $$ ...
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2answers
1k views

Two part question, Area of Torus using disk/ washer method

a. A torus is formed by revolving the region bounded by the circle $(x-2)^2 + y^2 = 1$ about the y-axis. Use the disk/washer method to calculate the volume of the torus. Figure given, showing $r=2$ ...
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1answer
64 views

elementary calculus inquiry

Show that under the transformation $x = \rho\cos\phi$, $y = \rho\sin\phi$, the equation $$ \frac{\partial^2u}{\partial x^2} = \frac{\partial^2u}{\partial y^2} = 0$$ becomes ...
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1answer
829 views

Global maximum/ minimum of a function of more than one variable.

Please, can someone give me more information on how to check if points are local or global maximum/minimum. I am aware of the second derivative test of determining the local minimum/maximum. But how ...
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2answers
2k views

Calculate the partial derivative, local minima/maxima, and saddle points.

I am having trouble finding the partial derivative. And clues or hints regarding said problem and how to find saddle points/local maxima is appreciated.
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1answer
277 views

Global Min-Max Optimization

When is \begin{equation} \min_X \max_Y f(X,Y) \end{equation} globally solvable? (i.e. we can find global solution for the optimization problem?) I am not looking for reformulations. Is it only when ...
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2answers
234 views

algebraic manipulation of differential form

Suppose $\phi_1, \phi_2, \dots, \phi_k \in (\mathbb{R}^n)^*$, and $\mathbf{v}_1, \dots, \mathbf{v}_k \in \mathbb{R}^n$ $(\mathbb{R}^n)^*$ stands for the space of all linear transformations that goes ...
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1answer
198 views

Conceptual question about area elements and volume elements

I have a couple of questions about area elements and volume elements and why they are the form they are when we transform between different coordinate systems. Say we have some curve parametrized in ...
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2answers
305 views

Proving multivariable non-differentiability using this particular definition

So I've been asked to prove non-differentiability using this particular method, and I'm a bit lost. I'm supposed to prove that $f(x,y) = |x-y|$ is not differentiable along the $x=y$ line. To do it, ...
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1answer
391 views

Using Divergence Theorem on boundary of surface

Use the Divergence Theorem to evaluate $$ \iint_{S} \vec{F} \cdot \hat{n}\,dS, $$ where $ \vec{F} = \langle 4x, 2y^2, z^2 \rangle, S $ is the boundary of the region defined by $ x^2 + y^2 + z^2 \leq ...
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1answer
109 views

Multiple Integration, polar coordinates

I'm having trouble with the following question: Find the area of the region inside both of the circles $r = 2a\cos(\theta)$ and $r = 2a\sin(\theta)$, where a is a positive constant. From what ...
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1answer
1k views

Spherical Coordinates Triple Integral

Write a triple integral in spherical coordinates that expresses the volume of the solid formed when a sphere with radius $a$ tangent to the $xy$ plane at the origin intersects at the plane z = a. ...
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1answer
62 views

What's the value of this partial derivative?

I need to find $\frac{\partial^2x}{\partial t^2}$, where $x = r\sin t$ and $y = r\cos t$. I get the following: $$\frac{\partial^2z}{\partial x^2} r^2\sin^2 t + 2\frac{\partial^2z}{\partial y\partial ...
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1answer
177 views

Inequality involving the integral of vector valued function

Suppose $f :[a,b]\to \mathbb{R}^n$ is given by $f(t)=\langle f_1(t), \ldots, f_n(t)\rangle$, where each of the $n$ component functions is integrable over $[a,b]$. I think the following inequality ...
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2answers
94 views

Show that $DF(x,y)$ is invertible in a dense and open subset of $\mathbb {R^2}$

My problem is the following: Let $p$ be a non constant polynomial over $\mathbb {R}$ and define $F(x,y)=(p(x+y),p(x-y))$.Show that $DF(x,y)$ is invertible in a dense and open subset of $\mathbb ...
3
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1answer
204 views

Saddle points - Show that surface $z = y \sin x$ has infinitely many saddle points.

Show that surface $z = y \sin x$ has infinitely many saddle points. Can someone show me the step-by-step solution for that statement? Detailed explanations will be appreciated. Thank you very much!
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1answer
1k views

Line integral vs Arc Length

I am trying to understand when do to line integral and when to do arc length. So I know the formula for arc length varies based on $dx$ or $dy$ like so: $s=\int_a^b \sqrt{1+[f'(x)]^2} \, \mathrm{d} x$ ...
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2answers
346 views

Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables

Problem 3-29 (p. 61) in the section treating Fubini´s theorem reads: Use Fubini´s theorem to derive an expression for the volume of a set of $\mathbb{R}^{3}$ obtained by revolving a Jordan-measurable ...
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1answer
519 views

Gauss divergence theorem

I'm trying to solve the following problem using the Gauss divergence theorem. I have to calculate the Flux through a sphere. The sphere is given as $$ x^2 +y^2+z^2==4 $$ where as the z is resticted ...
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0answers
64 views

Prove that the following differntial equation holds for given implicitly defined function

I'm given implicit function $z=F(\frac{x}{z},\frac{y}{z})$. For $z(x,y)$ I want to show that $$x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=z.$$ I'm using implicit function theorem. ...
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1answer
292 views

Calculus 3 Explained

I am trying to learn some calculus 3 and I understand HOW to do the problems but I just don't understand WHY I'm doing what I'm doing. So does anyone have any good recommendations on books that are ...
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3answers
93 views

Need help in solving volume integral

$$\int\int\int_{V}(x-y)dV$$ where $V$ is volume enclosed by : $$ S=\left\{(x,y,z):(x^{2}+y^{2})^{2}+z^{4}=16;z\geq0\right\}$$ What I did: ...
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2answers
50 views

Need help with this parametric equation

$s(t)=(\frac{2}{t^2+1},\frac{2t}{t^2+1})$ I need to calculate a line integral along this path. But I have trouble understanding what it is. I did some googling and it looks that it is a parabola, but ...
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1answer
2k views

Parametric equation of a cone

I usually use the following parametric equation to find the surface area of a regular cone $z=\sqrt{x^2+y^2}$: $$x=r\cos\theta$$ $$y=r\sin\theta$$ $$z=r$$ And make $0\leq r \leq 2\pi$, $0 \leq \theta ...
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1answer
925 views

How to find the boundary curve of a surface, like the Möbius strip?

I feel like I am missing a key piece of intuition in trying to understand this. I have just recently started using Stoke's theorem and I struggle to see what the boundary curve of surfaces are. In ...
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1answer
55 views

Is there a constant for this?

Suppose that $\sum_{i=1}^{n}\lambda_{i}=1$, where $\lambda_{i}>0$, and $\sum_{i=1}^{n}x_{i}^{2}=1$, where $x_{i}>0$. Does one have $n^{3/2}\min_{1\le i\le n}\lambda_{i}x_{i}\le B$ for some ...
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0answers
136 views

using stokes theorem to calculate a line integral [duplicate]

Use stokes theorem to show that: $$\int_c ydx + zdy +xdz = -\sqrt{3} \pi a^2$$ Where c is the suitably oriented intersection of the surfaces $x^2 + y^2 +z^2=a^2$ and the plane $x+y+z=0$.
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1answer
28 views

Projections on a disk in $\mathbb{R}^n$, Lagrange Multiplers

Suppose $S = \{x \in \mathbb{R}^n : \Vert x \Vert \leq 1\}$. Find the projection of a point to $S$, so find the shortest distance subject to the unit disk. That is we need to solve the following ...
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2answers
91 views

$\nabla \times X$ is given on a surface, can I show that X = 0 on the same surface

If I have a volume V enclosed by a surface S, and $\nabla \times X$ is given on the surface, what information does that give me about X on S. Is there a method of showing that X = 0 on S? (in the ...
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1answer
291 views

Directional Derivative

Given the following equation: $V(x,y,z)=5x^2-3xy+xyz$ Part 1: At point $P(3,4,5)$, find the rate of change in the direction of the vector $\langle1,1,-1\rangle$ Part 2: Find the direction in which ...
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2answers
2k views

How to verify a solution to the Wave Equation?

I have a wave equation: $$\frac{\partial^2u}{\partial t^2} = a^2 \frac{\partial^2u}{\partial x^2}.$$ How would I verify that the function $u(x,t)=\sin(x-at)$ satisfies the aforementioned wave ...
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1answer
59 views

First order differential equation

Im clueless on how to solve the following question... $xe^y\frac {dy}{dx} = e^y +1$ What i've done is... $\frac {dy}{dx} = \frac 1x + \frac {1}{xe^e}; \frac {dy}{dx} - \frac {1}{xe^e} = \frac 1x $ ...
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2answers
144 views

Evaluate line integral

I've been trying to work out this problem - the answer's $28$, but I can't understand how my textbook gets to that. I have Vector $\vec F(x,y,z) = yz^2\hat i + xz^2\hat j + 2xyz\hat k$ and $C$ is the ...
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1answer
241 views

Prove the functions are unique in a volume, vector calculus problem

I am working through the following problem, but finding it hard to know where to go. Using the Divergence theorem and the following identities $\nabla .(A \times B) = B.(\nabla \times A) - A.(\nabla ...
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1answer
1k views

Average distance from a point on a sphere to the origin

Consider a sphere $S$ of radius $a$ centered at the origin. Find the average distance between a point in the sphere to the origin. We know that the distance $d = \sqrt{x^2+y^2+z^2}$. If we ...
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2answers
75 views

how can I find maximum value of this function?

Can any one help me to calculate this function : $$f(y)=\max\limits_{\mu>0}[\exp(\frac{-n\mu^{2}}{\sigma^{2}})\exp(\frac{2\mu}{\sigma^{2}}\sum_{k=1}^{n}y_{k})]$$ where $y_{k}$ is random variable ...
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0answers
74 views

Help on a small proof of a theorem.

$\def\di\{\mbox{div}} \def\n{\nabla}$ I'd like help to undertand the small proof of the theorem 5.2 here (dowload). How can I see that \begin{equation} 0 = \int_{\{\xi > 0\}}\di \left \{ \Bigl( ...
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1answer
93 views

Taking a limit of $f(x,y)$ using the definition

I am reviewing for a test in my Calculus III class and I ran into a problem. I need to prove that $\lim_{(x,y)\to(a,b)} x+y=a+b$ using the formal definition of a limit. The definition given is: Let ...
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1answer
191 views

Applying Green's Theorem

So I'm practicing a few problems and I can't get this one - $$P(x,y) = e^x \sin(y) \\ Q(x,y) = e^x \cos(y)$$ $C$ is the right hand loop of the graph of the polar equation $r^2 = 4\cos(\theta)$ ...
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2answers
116 views

How does partial derivative work?

I don't understand the second step at all. Where did the $\partial^2 u/ \partial x^2$ come from and why do we have six terms?
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1answer
175 views

Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?

This is a follow-up to Continuous partials at a point but not differentiable there?, but I'll make this question self-contained. Throughout, $f$ will denote a function $\mathbb{R}^2\to\mathbb{R}$. An ...
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4answers
883 views

Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant. *help needed please*

Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant. This is what i've done so far.... Let $a,b,c$ be some points that the ...
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215 views

Jacobian and integration by substitution formula

Suppose $\int_S \nabla f(x) \cdot \nabla f(x)\;dx$ where $x = (x_1, ..., x_n)$. Substituting $x = g(y)$, where $g:T \to S$ is injective and $C^1$ and $y = (y_1, ..., y_n)$. So the integral becomes ...
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179 views

Does this hold?

Strayed on the following question. Assume that $x_{1}$,$\ldots$, $x_{d}\ge0$ with $x_{1}+\ldots+x_{d}=1$ and $y_{1},\ldots,y_{d}\in\mathbb{R}$. Does $$ \min_{1\le i\ne j\le ...
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1answer
151 views

Chain Rule for Vectorial Function

Let be $f:\mathbb{R}^N\rightarrow \mathbb{R}$. Let be $r$ a vector function, such that $r:\mathbb{R}\rightarrow \mathbb{R}^N$. Making $r(t)=y$ and $g(t)=f[r(t)]$. My lecture say: $g'(t)=\nabla ...
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1answer
585 views

Stokes' theorem - circle cross section in a sphere

A circle C is cut on the surface of the hemisphere $x^2 + y^2 + z^2 = 1,z≥0 $ by the cylinder $ x^2 + y^2 = y. $Evaluate $ \int_{C} -y^2\,dx +y^2\,dy+z^2\,dz $ where the direction round C is such that ...
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1answer
502 views

Continuous partials at a point but not differentiable there?

In Question on differentiability at a point, it is mentioned (and in Equivalent condition for differentiability on partial derivatives it is cited from Apostol) that for $f:\mathbb{R}^2\to\mathbb{R}$ ...