# Tagged Questions

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### Global Max and Min Problem

I'm working on a problem which asks me to find local and global extrema of the following function. $$f(x,y) = x^2y^2e^{(-x^2 - 2y^2)}$$ I went through and found all of the relevant partial ...
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### evaluating surface integral with divergence theory

If I have to calculate the surface integral of $\iint_S A \cdot n\ \mathrm {ds}$ where $A= 3zi-2xj+5x^2zk$ and $S$ is the surface of the cylinder $x^2+y^2=4$ and lying between $z=0$ and $z=4$ in the ...
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### Surface Integral of a Right Circular Cone

Use a surface integral to show that the surface area of a right circular cone of radius $R$ and height $h$ is $\pi R \sqrt{h^2+R^2}$. Hint -- Use the parametrization $x=r\cos\theta$, $y=r\sin\theta$, ...
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### Divergence Theorem Question

$$\iint\limits_\sum f \ d \sigma = \iiint\limits_S \operatorname{div} \textbf{f} \ dV$$ $$\operatorname{div} \textbf{f}=1+2+3=6$$ After this, we could multiply $6$ by the volume of the sphere ...
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### Lagrange multipliers (distance)

Find the closest point of the surface $z=xy-1$ to the origin. How would you do that with Lagrange multipliers?
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### Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
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### Find partial derivatives of $f(x, y)=\sqrt[3]{xy}$

Let $f(x, y)=\sqrt[3]{xy}$. Find $f_x(0,0)$ and $f_y(0,0)$. Is $f$ differential at (0,0)? How can I do?
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### Calculating $\iiint_K \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$.

I need to calculate the following in cylindrical coordinates: $$\iiint_K \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$$ $K$ is bounded by the plane $z=3$ and by the cone $x^2+y^2=z^2$. I know that: ...
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### Problems regarding multivariable calculus

Let $f:\Omega\to \mathbb R$ be differentiable at $x_0\in \Omega$ ($\Omega$ is a nonempty open subset of $\mathbb R^n$), let $f(x_0)=0$ and let $g:\Omega\to \mathbb R$ be continuous at $x_0$. We want ...
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### Newton's binomial for matrices that don't commute?

I'll give a bit of background info as to why I'm asking. I need to find the directional derivative of $f(A)=A^m$ where $m>0$ and $A$ is an $n$ by $n$ matrix with real entries. I want to do this ...
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### How to find the differential of this function

we are given the function $f: \mathbb R^n \setminus \{0\} \to \mathbb R^n$ defined by: $f(x) = \frac{x}{|x|}$ Find $Df(a)$. What I did: I tried working this out from the definition. the ...
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### a question about how to prove mutivariable integral, I am struggling about it!

If $f(x)$ is Riemann integrable in $[a,b]$, and then how to prove \int_{a}^{b} f(x_1) \, dx_1 \int_{a}^{x_1}f(x_2) \, dx_2 \cdots \int_{a}^{x_{n-1}}f(x_n) \, dx_n={1\over n!} \left[\int_a^b f(x) \, ...
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### Problem with a normal

Text of problem: "Define equation of curve, in all it points normal have the following feature: length of abscissa on the $x$-axis between beginning of coordinates and intersection of normal with ...
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### Verifying the Divergence Theorem for Half of a Sphere

Here is an exercise that I was assigned for homework: .......................................................... To the bottom left, I have scanned an example problem for verifying the divergence ...
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### Continuity of a function in two variables

Function $f(x,y)$ is continuous in each variable separately. Prove that there exists a point where it is continuous in two variables. I do not quite understand how to act here. I know the ...
A problem (among a list of Lagrange multipliers problems in Earl Swokowski's Calculus) states as follows: find the shortest distance between $2x+3y-z = 2$ and $2x+3y-z=4$. I can see that the ...
Prove the given formula. So far I have $f\textbf{F}=(f\textbf{F}_1, f\textbf{F}_2, f\textbf{F}_3)$, but I'm not sure where to go from there. Could anyone give me some pointers? Thank you.