# Tagged Questions

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### The 10th derivative of the f(x,y)= e^x * cosy in the neighborhood of (0,0).

I think this goes with the Taylor's formula in the point (0,0).. So, I have found first the Taylor's expansion of the function e^x than from the second one cosy and multiplied them..but I am not sure ...
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### Problem using chain rule

Let $f:\mathbb R \to \mathbb R$ a function of class $C^1$, and let $g(x,y)=f\left(\dfrac{x-y}{x+y}\right)$ for all $x \neq -y$ Prove that the direction of greatest increase of $g$ at $(x_0,y_0)$ ...
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### Maximum of the function of multivariable?

I need to find the maximum of the function given by $z=x^3+xy$ in $A=[0,1]\times[0,1]$. I think I need to use partial derivatives, but I'm not sure exactly how.
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### Proof of solid angle theorem

I have a homework problem to prove about the solid angle. The book says: Let S be a smooth parametric surface and let P be a point such that each line that starts at P intersects S at most once. ...
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### Knowing when to use Green/Stokes/Divergence theorem to evaluate line/surface integrals

$\newcommand{\mbf}{\mathbf}$ Evaluate $$\iint \limits_{S} \mbf{F} \cdot d \mbf{S}$$ where $\mbf{F} = 3xy^2 \mbf{i} + 3x^2y \mbf{j} + z^3 \mbf{k}$ and $S$ is the surface of the unit sphere. I ...
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### What is $\lim_{(x,y)\to(2,2)}\frac{x^4-y^4}{x^2 - y^2}$?

I have limit: $$\lim_{(x,y)\to(2,2)}\frac{x^4-y^4}{x^2 - y^2}$$ Why is the result $8$ ?
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### Question on Curl F

The problem in the book asks what the curl of $\operatorname{curl}\vec F(\vec r)= \frac {\vec r}{\|\vec r\|}$. Can someone give me a good explanation on why the curl will be zero? I would really ...
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### Marginal density function understanding

Given a plane with three points, $(0, -1)$, $(2,0)$, and $(0, 1)$ with $x$-axis and $y$-axis connecting three points to make a triangle. Suppose this triangle represents the support for a joint ...
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### How to find the partial derivative of this function?

Lets say I have a function:$$\nu=\frac{RT}{P}+B_{p}(T)RT$$ and I am trying to find $(\frac{\partial \nu}{\partial T})_{P}$. I understanding that the partial derivative of the first term is just ...
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### Rigorous proof of the “Lagrange-multiplier theorem”

From Marsden's Elementary Classical Analysis: Theorem 8 Let $f\colon U \subset \Bbb R^n \to \Bbb R$ and $g\colon U\subset \Bbb R^n \to R$ be given $C^1$ functions. Let $x_0\in U$, ...
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### Help with Stokes problem

Well, I hope this is a stokes problem. Im honestly a bit lost on this so please help me out! Suppose I have a simple closed curve, C, in the plane w/ counterclockwise direction. I need to calculate ...
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### The differentiability class of the inverse function

Here's the final part of a proof (from Marden's Elementary Classical Analysis) of the inverse function theorem, where we have been given that $f$ is of class $C^p$: Could someone please explain the ...
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### Calculating volume bounded by area

The problem is calculation of the volume bounded by $(x^2+y^2+z^2)^2=\frac{64}{x^2+y^2}$ I tried using cylindrical coordinates: $x=r*cos(t) ; y=r*sin(t); z=z$ then this gives me: ...
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### Inverting a Coordinate Transformation

Given $x(s,t)=s+t$ and $y(s,t)=st$ is it possible to find $s$ and $t$ as functions of $x$ and $y$? I'm given a function $f(x,y)=x^2+xy+y^2$ and told it is expressed as $h(s,t)$ where $x(s,t)=s+t$ and ...
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### Question about proof of Morse Lemma

I am working on a problem for my differential geometry course. We are proving the following special case of the Morse lemma: Let $U \subseteq \mathbb{R}^n$ be open and containing the origin ...
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### Differential and partial derivates

Can you help with the following question? Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a function with continuous derivatives. It is given that $f\left(\dfrac{\cos(t)}{t},\dfrac{\sin(t)}{t}\right)$ is ...
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Temperature of the point $(x,y,z)$ in space is measured by the formula $T(x,y,z) = e^{-x^{2}-2y^{2}-3z^{2}}$ . At the present an object is located at the point (1,1,1) and needs to be brought to a ...
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### Manifolds - 1D distribution on $S^2$

I'm trying to find a one dimensional smooth distribution on $\mathbb{S}^2$ - i.e. a map $p \mapsto D_p$, where $D_p$ is a one dimensional subspace of $T_p\mathbb{S}^2$, so that $D_p$ is locally ...
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### how to prove continuity in multivariable functions?

$$\frac{x^3y}{x^4+y^2},etc.,$$ in these multi-variable functions its easy to prove discontinuity by giving counterexamples but for proving continuity are there any tricks? using $\epsilon$ ...
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### An strange triple integral I've never seen before

This is one of the integrals I have for homework, But I've never seen anything like this before, I don't know what to do with it. Does anyone know? $$\int_0^1 dz \int_z^1 dx \int_0^x e^{x^2} dy.$$ ...
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### find fx in terms of ∂w/∂r and ∂w/∂β

x=rcosβ y=rsinβ w=f(x,y) ∂w/∂r = fxcosβ + fysinβ and (1/r)*(∂w/∂r)= -fxsinβ + fycosβ Express fx and fy in terms of ∂w/∂r and ∂w/∂β. What I've got so far fy= (∂w/∂r)(1/sinβ)-fx(cosβ/sinβ) and ...
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### Find the intersection of the surface $z=2x^2+y^2$ with the $x-y$ plane

I was asked to find the partial derivatives of $z=2x^2+y^2$ and the intersection of this surface with the $x-y$ plane. Is finding it relates to the partial derivatives? Here are my solutions for the ...
Let $P$ be a plane in $\mathbb{R}^3$ parallel to the $xy$-plane. Let $\Omega$ be a closed, bounded set in the $xy$-plane with $2$-volume $B$. Pick a point $Q$ in $P$ and make a pyramid by joining ...
### Help with differentiable function $f:\mathbb{R^2} \to \mathbb{R}$
I need help with a question that appeared in my test. True or False: Let $f$ be a function $f:\mathbb{R^2} \to \mathbb{R}$ not differentiable at (0,0), then $f^2$ is not differentiable at (0,0). ...