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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30 views

Help With Understanding a Specific Derivative

I'm working through the MIT 18.02 2007 Multivariate Calc class right now, and I don't understand a derivative they use in their answer key. Specifically, given $T = \frac{-a\sin tI +a\cos tJ + ...
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61 views

Exercise on Jacobians/Triple Integrals

Q: Integrate the function $f=x^2+y^2$ over an elliptical cone, with the base being the ellipse $\frac{x^2}{4}+y^2=1$, $z=0$ and the apex at the point $(0,0,5)$. The integral can be made simpler by ...
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1answer
36 views

Simplify curl$(\mathbf{F} \times \mathbf{G})$ under these hypotheses on $F, G$ - 2011 4C

Hypotheses: $F$ is a constant vector $\nabla \times F = \mathbf{ 0 }$ $\nabla \cdot F = 0 $ $\nabla \cdot G = 0 $ From my other post, $[\color{green}{\nabla} \color{brown}{\times} (\mathbf{F} \times ...
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1answer
100 views

Convexity of a rational function

I am attempting to (dis)prove that the function $$\frac{4x+3y+2}{x^2+xy+2x+y}$$ is convex for $x,y>0$. Attempting to differentiate the function does not seem like a good idea (or am I making a ...
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0answers
49 views

Brouwer's fixed theorem using Stokes' theorem

according to Wikipedia, there is a simple way to prove Brouwer's fixed point theorem from Stokes' theorem: see here. So I would like to present the former famous theorem (Brouwer's one) to my Calculus ...
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3answers
97 views

Convexity of $\sqrt{x^2+y^2}$

I am to prove that $\sqrt{x^2+y^2}$ is convex for $x,y>0$. Intuitively, if I look at the derivatives, $\frac{x}{\sqrt{x^2+y^2}}$, $\frac{y}{\sqrt{x^2+y^2}}$, they are increasing in every positive ...
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1answer
46 views

Double integral inequality

Proof this inequality: $$\int_1^4 \int_0^1 (x^2+\sqrt{y})\cos(x^2y^2) dx dy\leq 9 $$ I don't know how to approach to this, any idea ?
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1answer
123 views

why normalize and the definition of directional derivative

I'm not understanding how to solve this problem. I think the problem lies in the fact that I don't understand why normalize and the definition of directional derivative... It's in Portuguese. The ...
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2answers
99 views

Implicit function theorem problem

I have the function $$(x-2)^3y+xe^{y-1}=0$$ And I have to see if $y$ can be described as a function of $x$ around (1,1). The implicit function theorem can't be applied in this case. What should I ...
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1answer
37 views

Line Integral and potential Fields

Field : $$\int_\ell \frac{-1}{1+(y-x)^2}\,dx + \frac{1}{1+(y-x)^2}\,dy$$ Find the path from point $(0,0)$ to $(1,2)$ along the ellipse $(x-1)^2 +(y/2)^2 =1$. I thought of checking the green formula ...
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50 views

A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
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3answers
249 views

Verifying Stokes' Theorem

Verify Stokes' Theorem for the given vector field $f(x, y, z)$ and surface $\Sigma$. $$f(x, y, z) = 2y \textbf{i} - x \textbf{j} + z \textbf{k}; \quad \Sigma : x^2 + y^2 + z^2 = 1, z \ge 0$$ This was ...
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2answers
20 views

Prove set is bounded and nonempty?

Suppose I have a function $f(x, y) = 1-x^2-y^2$ and I have the set $A = f^{-1}([C, \infty)$. I need to show that it is compact. I have already shown that it is closed but I'm not sure how to show it ...
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1answer
54 views

Partial derivative of $g(x,y)=f(h(x,y),l(x,y))$

Let $f,h,l: \mathbb{R}^2 \to \mathbb{R}$ be derivable functions. If $g(x,y)=f(h(x,y),l(x,y))$, is the following formula true? $$\frac{∂g}{∂x}(x,y)= \left(\frac{∂f}{∂x}(h(x,y),l(x,y)) ...
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0answers
128 views

Taking the improper integral of a series term-by-term.

Suppose I have a function $f$ which appears in some complicated formula in a term that looks like $\int_{-\infty}^{\infty} f(x, v)\,\text{d}v$. Basically, I want to write $f$ in a series of functions ...
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2answers
127 views

$\int_{0}^{\infty}\int_{1}^{\infty}\frac{x^2-y^2}{(x^2+y^2)^2}dxdy$ diverges?

I want someone to review my proof that $$\int_{1}^{\infty}\int_{0}^{\infty}\frac{x^2-y^2}{(x^2+y^2)^2}dxdy$$ does not converge. To make things easier, I said let's look at the entire first quadrant ...
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1answer
46 views

Limits in two variables. Basic property

I do not remember (1) the name for the following property and (2) the conditions that $f$ must meet in order to apply it, apart from being continuous and supposing that the left-hand-side limit ...
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1answer
29 views

What is an intuitive extension of extreme-values and critical points in one variable to multiple variables?

While it is simple to grasp limits in multiple variables, since the formal definition extends in the obvious way, I am having a harder time grasping the same concept with critical points and extreme ...
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1answer
18 views

Representing a triple integral in a different order of integration

I am given with the following question: A) $V_1 = \{ x^2 + y^2 \leq 4 , 0\leq z\leq 3 \sqrt{x^2 + y^2 } , x\geq 0 \} $ , and I need to represent the triple integral $\int \int \int_{V_1} f(z) dxdydz$ ...
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1answer
38 views

Line Integral on polygonal path

How would you calculate this? Since $\textbf{F}$ has to be parametrized, could $(x=t)$ be used? And what about $y$ and $z$? Or, since the curve is a closed path, is it automatically zero? ...
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1answer
56 views

Evaluate double integral:

Evaluate: $$\iint_{D} \arcsin(x^2+y^2)\, dx\,dy$$ where $D$ is defined by the following polar equation $\rho=\sqrt{\sin \theta}$ and $0\le\theta\le\pi$
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1answer
105 views

curve integral - intersection between plane and sphere

I am going to calculate the line integral $$ \int_\gamma z^4dx+x^2dy+y^8dz,$$ where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation ...
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3answers
163 views

$\int_{0}^{1}\int_{0}^{1}\frac{1}{\sqrt{x^2+y^2}}\,dx\,dy=$?

I'm having difficulties thinking of a good variable change for $$\int_0^1 \int_0^1 \frac{1}{\sqrt{x^2+y^2}}\,dx\,dy=?$$ the most natural choice would be something like $x=r\cos\theta$ and ...
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5answers
237 views

Prove that the gradient of a unit vector equals 2/magnitude of the vector

Let $\vec r=(x,y,z)$ Firstly find $\vec \nabla (\frac 1 r)$ where r is the magnitude of $\vec r$. I think I've done this correctly to get $-x(x^2+y^2+z^2)^{-\frac32} \hat i-y(x^2+y^2+z^2)^{-\frac32} ...
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0answers
70 views

How to derive multi-variable function $z=\sin(\sqrt{u^2-r})$, where $u=\sin x$ and $ r=e^x $?

How to derive multi-variable function $z=\sin(\sqrt{u^2-r})$, where $u=\sin x$ and $ r=e^x $? I have been given task to derive this function by using "the law of derivation of two variable composite ...
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1answer
30 views

Does every function with $f_x,f_y>0,f_{xx},f_{yy}<0$ with particular condition have to satisfy $f_{xy}/f_{xx} = -x/y$?

Let continuous real functions $f$ of two real variables $x,y$ satisfy the following condition: (Let us define $f_{xx}:=\frac{\partial^2 f}{\partial x^2}$ and $f_x:=\frac{\partial f}{\partial x}$, and ...
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1answer
30 views

Stokes' Theorem - How to find the surface?

Calculate $\int_C \textbf{f} \ d \textbf{r}$ for given vector field and curve $C$. $\textbf{f}(x,y,z)=\textbf{i}-\textbf{j}+\textbf{k}$ $C: x=3t, y=2t, z=t, 0 \leq t \leq 1$ Using Stokes' Theorem, ...
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2answers
714 views

Constructing a Cone and its Normal Vectors in Spherical Coordinates

I am attempting to construct a right circular cone of maximum radius $R$ and angle $\theta$ in spherical coordinates, then find the normal vector of the surface of this cone at all points. Here's what ...
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0answers
29 views

Characterization of the derivative as a tensor field

I was thinking about the derivative, and I wanted to make sure I’m thinking about it the right way. Suppose we have a $C^{\infty}$ function $f: {V}\to \mathbb{R}$, where $V$ is a finite-dimensional ...
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1answer
48 views

Explain the minus sign in the following formula.

I just read that: If $z=f(x,y)=c$, be the equation of a curve, then the slope of the tangent to the curve at any point (x,y), is given by $$m=\frac {dy}{dx}=-\frac{\frac{\partial z}{\partial ...
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0answers
21 views

How to show that the variance increases with the dimension $n$?

This can be seen as a statistics related question, but it is actually a more general mathematics related one. I am trying to understand the Particle Filter and the motivation to use it over the ...
2
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1answer
159 views

Prove equality of two vectors if they have equal divergence and equal curls

I have following question: Fields with equal divergence and equal curls $F_1$ and $F_2$ are two vectors fields, you may write them as $F_1 = M_1i+N_1j+P_1k$, $F_2 = M_2i+N_2j+P_2k$. Suppose that ...
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1answer
37 views

Find the volume a solid by triple integration

We need to find the volume of the solid bounded by the $xy$-plane, the cylinder $x^2+y^2=2x$, and the cone $z=\sqrt{x^2+y^2}.$ The volume is given by the triple integral $\int \int \int_S ...
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1answer
30 views

one problem on multivariable claculus

Suppose $\phi(\bar{x}(t))$ be a function which takes vectors (parameterized by $t$) as argument. Now take $c$ be a minimum point of the function $\phi$. consider a curve $\gamma(t)$ which passes ...
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1answer
27 views

Help Understanding Evaluation of Integral

Please help me to understand the evaluation of this integral. $$\int_0^1\int_u^{\mathrm{min(1,u+z)}} 2\;dv\;du$$ I know that the correct answer is $$ f(z) = \left\{ \begin{array}{lr} ...
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3answers
480 views

Line Integrals and Surface Integrals

Can someone please explain what surface integrals and line integrals are measuring? Is a line integral the arc length along a surface, and a surface integral is the surface area? Also, why is a line ...
2
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1answer
81 views

$\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ check my answer!

I would like someone to review my solution please, the original question is to calculate $\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ What I did: First I changed variables ...
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1answer
37 views

Divergence Computation in Gauge Theories, Knots and Gravity

Hopefully this is just some minor confusion...The first exercise wants us to show that $$\vec \epsilon(t,\vec x)=\vec Ee^{-i(wt-\vec k \cdot\vec x )}$$ satisfies the vacuum Maxwell equations where ...
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1answer
121 views

Fourier's Heat Law In Integral Form

I am having a little trouble with something. Here is a link (wikipedia article) to Fourier's Heat Law in integral form: http://en.wikipedia.org/wiki/Thermal_conduction#Integral_form What I am trying ...
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1answer
32 views

Proof Green's theorem $F(x,y)=(x-y)i+xj$

I was reading on Green's theorem and have appreciated the concept. Given a question, I think, I can solve it.But I came across a question that reads: Verify the Green's theorem for the vector given ...
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0answers
29 views

Sign of a two variable function

How to represent in a graph the sign of a two variable function? For example: $f(x,y)=x^2y$
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1answer
97 views

Clarifying definition of outward unit normal

I would like to figure out how to properly define the outward unit normal vector $\nu$ for a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$ ($n \ge 2$). I am ...
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2answers
54 views

How to Complete Sketch of a function of two variables $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ ? [Stewart P930 Question 14.7.4]

For $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ $\implies$ $\partial_x f = 3 - 3x^2, \partial_y f = -4y + 4y^3$. Set both equations to 0 $\implies x = \pm $1 and $y = 0, \pm 1$. $1.$ To determine the ...
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1answer
120 views

Can anyone help me with these double Integrals using mathematica

$$ \int_0^6 \int_0^4 \frac{\sqrt{(1+x^2+y^2 )^2+4 ( x^2+y^2)}}{1+x^2+y^2}\, dy \, dx$$ And $$\int_{-1}^1 \int_{-y}^y \frac{1}{(1+y^2)^2} \sqrt{(1+y^2)^4 + 4x^2(1+y^2)^2 + 4y^2(1+x^2)^2} \, dx \, dy$$ ...
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1answer
128 views

Sketch Saddle Point of a function of two variables $ f(x, y) = 4 + x^3 + y^3 - 3xy$ [Stewart P930 Question 14.7.3]

As regards $ f(x, y) = 4 + x^3 + y^3 - 3xy$, I computed that (0,0) is a saddle point, and (1,1) is a local minimum. So I'm not asking about this, and am asking only about sketching contours. $1.$ ...
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1answer
31 views

The gradient in different dimensions

I study to final exam in calc 3. Question: Are my thoughts about the gradient correct? The gradient is a normal vector to a plane given a point in $xyz$-plane. With this vector you can calculate ...
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2answers
125 views

Verify Gauss’s Divergence Theorem

I have this assignment which we have not tackled and am getting mixed up in the divergence theorem tutorials like this one ...
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1answer
73 views

Show that total curvature of ellipse is $2\pi$

I'm trying to show that the total curvature $$K=\int_C\kappa\,ds$$ is equal to $2\pi$ over the ellipse $C$ with axes $a,b$ (and $\kappa$ is curvature). I computed: $$x(t)=(a\cos t,b\sin t,0) \\ ...
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0answers
43 views

Conditions of the vector-calculus Green's theorem

While studying the Green's theorem, I think a lot about whether there is an abundance of the condition (now called $X$) of $X:\quad$ $P$ and $Q$ (in the Wikipedia article, $L$ and $M$) have ...
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2answers
35 views

How would I finish this continuity proof?

I have a multivariable function $f$ with $$f(x, y) = \begin{cases} \frac{x^2+y^2}{y} & \text{if }y \neq 0\\ 0 & \text{if }y = 0 \end{cases}$$ and want to show that it is continuous at $(0, ...