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
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
484 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 ...
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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
98 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, ...
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1answer
72 views

If first 1 by 1 upper left submatrix (principal minor) = 0, conclude straightaway saddle point ? - Question 8

Find all local extremal points for the function $f(x,y) = x^3 - 3xy+y^3 $ and classify their type. For $H(f)(0,0),$ I see that $D_1 = \det [0] = 0$. So according to the criteria that I already posted ...
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1answer
56 views

For $f(x,y,z,\ w)=x^{5}+xy^{2}-zw$, how is this stationary point $\;$ a saddle point? - Question 14

14. a$)$ Find all stationary points of $f(x,y,z,\ w)=x^{5}+xy^{2}-zw$. $b)$ Classify the stationary points of $f$ as local maxima, local minima or saddle points. Provided Solution a $)$ We compute ...
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1answer
48 views

Proving that $\Delta f =0$

Let $\|\cdot\|$ denote the $\mathbb R^2$euclidean norm. Let a such that $\|\mathbf a\|=1$ Let $B$ denote the open unit disk in $\mathbb R^2$ Define $\displaystyle ...
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2answers
99 views

How to prove that $f(x,y)=3+2x+y$ is continuous?

The question is to prove that the function $f(x,y,z) = 3+2x+y$ is continuous everywhere. My approach uses the delta-epsilon method. $|(x,y)-(a,b)|\lt \delta$ then $|f(x,y)-f(a,b)|$. All I did was ...
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1answer
46 views

Limits of integration, double integral

$$\int_Y y\sqrt z\sqrt{4x^2+4y^2+1}dS$$ $Y$ is given by $z=x^2 + y^2$ and $x\leq0$, $y\leq0$, $1\leq x^2+y^2 \leq 9$ I'm having a bit of trouble with this one, not sure if I'm getting the limits ...
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0answers
29 views

A Hard integral in 2-D.

I'm having a trouble integrating (in $\mathbb{R}^2$) the following formula: $$\frac{t}{|B(x,t)|}\int_{B(x,t)} \frac{||y||}{(t-||x-y||^2)^{\frac{1}{2}}} dy $$ where $B(x,t)$ is the ball with center ...
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1answer
94 views

Double integral involving zeta function: $\int_0^\infty \frac{1-12y^2}{(1+4y^2)^3}\int_{1/2}^{\infty}\log|\zeta(x+iy)|~dx ~dy.$

I'm having trouble evaluating the following double integral: $$\int_0^\infty \frac{1-12y^2}{(1+4y^2)^3}\int_{1/2}^{\infty}\log|\zeta(x+iy)|~dx ~dy.$$ Do please remark that $\zeta$ is the zeta ...
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0answers
24 views

Some non-elementary types of two-variable function

Let $f$ be a two-variable real-valued function on disk $D \subset \mathbb{R^2}$ and $(a,b)$ is the center of $D$. $\text{}$ The first problem is about continuity and partial differentiability. At ...
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10answers
434 views

Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?
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1answer
40 views

Vector calculus, divergence theorem

Please help!!! Demonstrate the identity $\int_D \nabla \times \mathbf{F} dV = \oint_{\partial D} \mathbf{\hat{n}} \times \mathbf{F} dS$, by evaluating both sides of the equality for $\mathbf{F} = ...
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1answer
16 views

Surface integral check

If $S$ is a sphere with radius $r$ and centre $(a,b,c)$, and I want the surface integral of: $$\mathbf{F}=\begin{pmatrix} x\\y\\z\end{pmatrix}$$ over S, is this correct: ...
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2answers
64 views

How to integrate $e^{-z}$ over the ball $x^2 + y^2 + z^2 \leq 1$

I can't figure out how to obtain the limits for this triple integral since I can't visualize how $e^{-z}$ varies over a sphere in my head.
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1answer
39 views

Definition of an integral over a domain.

In calculus we generally use this notion: $\int_D f(x)\,dx$. I understand that when $D$ is an interval from $a$ to $b$ the integral is equivalent to $\lim \limits_{\|\Delta x\| \to 0} \sum_{i} f(x_i) ...
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1answer
93 views

How to integrate $\int_{0}^{1} \int_{0}^{\pi} \int_{0}^{\pi} r^2 \sin\theta \sqrt{1 - r^2\cos^2\theta - r^2\sin^2\theta} \,d\phi\, d\theta \,dr$

Find the center of mass of the hemispherical region $W$ defined by the inequalities $x^2 + y^2 + z^2 \leq 1$ and $z \geq 0$ with unity density. By symmetry we know the $x$ and $y$ coordinates ...
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2answers
43 views

simple question about $\nabla r$

In my physics notes, it says $\nabla r = \underline{e_r} = \frac{\underline{r}}{r}$ and $\nabla \frac{1}{r} = - \frac{\underline{r}}{r^3} = - \frac{1}{r^2} \underline{e_r}$ I don't quite ...
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0answers
96 views

Solving a system of integral-partial differential equations

Hi I am a student in electrical engineering. Currently I am facing a difficult problem solving a coupled integral-differential partial equations arising from mean field game. The problem is similar as ...
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5answers
104 views

How to think when solving $3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$?

Solve this differential equation $$3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$$ Usually, when we get these problems, they tell us what variable change is smart to do and we just ...
2
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1answer
80 views

Line integral: $u = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, z)$

Let $u = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, z)$ and D the domain bounded by the torus obtained by rotating the circunference $(x-2)^2 + z^2 =1, y=0$ around the z-axis. Show that $rot( u )=0 ...
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1answer
37 views

Optimization of a Sum of Variables

Let there be variables $A$, $B$, $C$, $D$, and $E$ such that a total of $N$ points is allocated among the variables: $A$+$B$+$C$+$D$+$E$=$N$, $N$∈$ℝ$. Let the corresponding point values returned by ...
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0answers
41 views

Can you explain this partial derivative approximation?

How do they go from the left side to the approximation on the right? What does $j$ mean in this summation? $$ {\partial f_i^n \over \partial t} \approx \sum_j {\partial f_i^n \over \partial \hat x_j} ...
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4answers
125 views

Double integral for $\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$

I'm trying to evaluate this $$\int_{0}^{1} \int_{-1}^{0} \frac {xy}{x^2 + y^2 + 1}\ dy\ dx$$ tried substition $$ u = {(x^2+y^2+1)}^{-1} \ \ du = \ln {(x^2+y^2+1)}$$ but du is not found in the ...
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1answer
62 views

Applied problem of multivariable calculus with integrals.

How to solve this problem: A medal has the shape of the portion of the plane $x + z = 1$ lying inside the cylinder $$x^2 + \frac{y^2}{2} = 1.$$ The edge of the medal is covered in cold and it costs ...
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2answers
109 views

How to find the limits of a triple integral converted to spherical coordinates

Find the integration limits of $\int_{0}^{3} \int_{0}^{\sqrt{9 - x^2}} \int_{0}^{\sqrt{9 - x^2 - y^2}} \frac{\sqrt{x^2 + y^2 + z^2}}{1 + (x^2 + y^2 + z^2)^2} dz dy dx$ in spherical coordinates. ...
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1answer
46 views

How do I find the limits for $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$?

Evaluate $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$ where $W$ is the solid bounded by the two spheres $x^2 + y^2 + z^2 = a^2$ and $x^2 + y^2 + z^2 = b^2$ where $0 < b < a$. ...
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1answer
74 views

Show $T(u,v,w) = (u \cos v \cos w, u \sin v \cos w, u \sin w)$ is onto the unit sphere

$T: \mathbb{R^3} \rightarrow \mathbb{R^3}$ is defined by $T(u,v,w) = (u \cos v \cos w, u \sin v \cos w, u \sin w)$. Show that $T$ is onto the unit sphere, $x^2 + y^2 + z^2 = 1$. I believe I have ...
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0answers
30 views

Gradient; how to do this?

I want to do this gradient, but I just don't get the right result: $\phi: \mathbb{R}^3 \rightarrow \mathbb{R}$ and $F(Y) = - q \ \text{grad}\phi(Y) = \frac{1}{4 \pi \varepsilon_0} ...
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0answers
58 views

How to solve Max under an integral?

This is the first time I come accross a Max function inside an integral. I have looked around online and did not find anything about it. I would like to know the rules of what can I do when I have an ...
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2answers
38 views

Evalulate $\iint_{D}^{} (x+y) dx dy$

Let $D$ be the region $0 \leq y \leq x$ and $0 \leq x \leq 1$. Evalulate $\iint_{D}^{} (x+y) dx dy$ by making the change of variables $x = u + v$ and $y = u - v$. ($D$ here is the image set, so ...
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2answers
19 views

A substitution that will simplify this integrand

$$\iint_{R}^{} x \sin(6x + 7y) - 3y \sin(6x + 7y) dA$$ So I chose $u = 3y$ and $v = 6x + 7y$. So then $x$ will be replaced with $\frac{3u - 7v}{18}$. It seems correct, but does this truly "simplify" ...
3
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1answer
1k views

cross product in cylindrical coordinates

Hi i know this is a really really simple question but it has me confused. I want to calculate the cross product of two vectors $$ \vec a \times \vec r. $$ The vectors are given by $$ \vec a= a\hat ...
2
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0answers
52 views

Finding flux of a vector field $\mathbf{F}$ across a surface bounded by an unknown function.

A solid $\Gamma$ in $\mathbb{R}^{3}$ is bounded by $$0 \leq x \leq 1, \hspace{.5cm} 0 \leq y \leq 1, \hspace{.5cm} 0 \leq z \leq g(x, y),$$ where $z = g(x, y)$ is an unknown differentiable surface. ...
3
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1answer
66 views

What must a function satisfy in order to say that a certain limit exists?

Suppose that $f:R_+\to R_+$ is $C^2$, and that $f(0)=f'(0)=f''(0)=0$, and that for all $x$, $\frac{xf''(x)}{f'(x)}\geq1$. From here we can safely say that $\lim\inf_{x\to0}\frac{xf''(x)}{f'(x)}\geq1$. ...
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2answers
64 views

Find the surface area of portion of the plane that is inside the cylinder.

I am given the plane $x + y + z = 1$ and the cylinder $x^2 + y^2 = 4,$ and have to find the surface area of portion of the plane that is inside the cylinder. I am very confused with this. I tried ...