# Tagged Questions

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### Partial derivative and derivative.

I want to show that if $f:\mathbb{R}^n\to \mathbb{R}$ and $df_a$ is the derivative of the function at $a$ then $df_a(v)=\displaystyle\frac{\partial f}{\partial v}(a)$. I saw a few proofs of this ...
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### Limits of norms and deriviative as linear transformation

I'm self-studying Spivak's Calculus on Manifolds and he introduces the derivative by first looking at it as a linear transformation, $Df(a) = \lambda$, saying that for a differentiable function ...
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### Find the volume of a cone whose length of its side is $R$

How can i compute the volume of a cone whose length of its side is $R$ and the vertex of the cone forms an angle $2θ$ . The top cone is a cap of a sphere of radius $R$. I tried to solve first in 2 ...
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### How do I convert the limit definition of differentiability to different variables?

I want to convert this: $$\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0) - J(h)}{\|h\|} = 0$$ Into the version of the limit where it has $\lim\limits_{(x,y) \to (0,0)}$ instead of $h$. How do I do this?
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### Does the existence of partial derivatives imply Frechet differentiability?

Let $f : \mathbb R^n \rightarrow \mathbb R^m$ and $a \in \mathbb R^n$such that $\forall i \in [1,n], \large \frac{\partial f}{\partial x_i}(a)$ exists. Is $f$ Frechet differentiable ? I'd say no, ...
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### What is the difference between $\Delta r$ and $dr$ in Taylor series

All I know about Taylor series is at here. It tells how to expand a funcion to a polynomial. However I see the Taylor series at the form like this (here $r$ is a parametrized surface of $u,v$): ...
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### Chain rule and multivariable derivatives

Given $F\colon\mathbb{R}^2\to\mathbb{R}$ defined as $F(x,y) = x^2 y + y^3 + 2x-1$ and $g\colon\mathbb{R}\to\mathbb{R}$ such that $g(0) = 1$ and $F(x, g(x)) = x$, find $g'(0)$.
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### Is finding the tangent plane to a surface made any more complicated if the surface $\neq 0$?

So I have $x^2 + y^2 - z^2 = 4$ as my surface and the point I'm looking at is $(2,1,1)$. So if it was $0$, I'd do my partial derivatives and get the equation: $4(x - 2 ) + 2(y-1) - 2(z-1) = 0$ ...
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### Multivariate integration of a derivative w.r.t. a single variable

$x=(x_{1},...,x_{n})$. If $\frac{\partial g(x)}{\partial x_{l}}=f(x_{l})$ for $l=1,...,n$, should we have $g(x)=\sum_{l=1}^{n}\int f(x_{l})dx_{l}+c$? If yes, what's the theorem or proposition behind ...
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### Locate and classify stationary points

Locate and classify as maxima, minima or saddle point the stationary points of the surface given by the equation $$z=(5x+7y-25)e^{-(x^2+xy+y^2)}.$$ Stationary points are the points where the gradient ...
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### Jacobian of mapping

Let's say we're in $\mathbb{R}^n \times \mathbb{R}^n$ and we have the identity mapping $f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n \times \mathbb{R}^n$, $f(x,y) = (x,y)$. What I want ...
### Partial derivatives of $f(x,y)=\sqrt{|xy|}$
$f(x,y)=\sqrt{|xy|}$ First question: How to find $f_x(0,0)$ and $f_y(0,0)$? I have figured out this using definition - Both are $0$. My next question is: How to show that $f_x(0,0)$ and $f_y(0,0)$ ...
Let $f \left( \begin{array}{ccc} x \\ y \end{array} \right)= \begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2} & \mbox{if$(x,y) \neq (0,0)$}\\ 0 & \mbox{if$(x,y) = (0,0)$} \end{cases}.$ I ...