2
votes
1answer
28 views

Prove that $f$ is differentiable in $(0,0)$ if and only if $\lim_{t\to0+} g(t)$ exists

Let $g:[0,\infty)\to\mathbb{R}$ be a mapping and $f(x,y)=xg(\sqrt{x^2+y^2})$ for all $(x,y)\in\mathbb{R^2}$. Prove that $f$ is differentiable in $(0,0)$ $\iff$ $\lim_{t\to0+} g(t)$ exists. My ...
1
vote
0answers
55 views

Verification: Hessian of the following composition.

I was hoping that someone could verify the steps of computing a Hessian matrix. I have the following function, $F:\mathbb{R}^n\to\mathbb{R}$, $$F({\bf x}) = \sum_{i=1}^mf(g(A_i^T{\bf x}))$$ where ...
0
votes
1answer
32 views

Using the implicit function theorem to solve for two of four variables in the system of two equations

Show that there are positive numbers $p$ and $q$ and unique functions $u$ and $v$ from the interval $(-1-p, -1+p)$ into the interval $(1-q, 1+q)$ satisfying $$xe^{u(x)} +u(x)e^{v(x)}=0=xe^{v(x)} ...
1
vote
1answer
27 views

Multivariable-calculus, derivative and second derivative [closed]

I got the function $f(x,y)=\ln \sqrt{x^2+y^2}$. The task is to find the derivative function and the second derivative function. How do I get there?
0
votes
1answer
66 views

Two methods of finding a function $f$ such that $Mdx+Ndy=0$ on the curves $f(x,y)=c$

this problem is from my class,i did one way and got one answer,professor did it in another way and got another answer.question is:Find $f(x,y)=constant$ where differential equation is ...
2
votes
2answers
40 views

Direction of Greatest Increase

Problem: Find the direction of greatest increase at $P$. $$f(x,y)=4x^2+y^2+2y$$ $$P=(1,2,12)$$ Solution: The greatest increase in $f(x,y)$ at $P$ can be attained by moving in the direction of ...
4
votes
1answer
40 views

Derivative: $f_x, f_y, f_{xy}$ of function - $f(x,y)$

Let's say $f(x,y) = x^2 + 2xy +y^2$ $f'_x = 2x + 2y$ $f'_y = 2y + 2x$ $f'_{xy} = 2x + 2y$ ? Am I right about the third?
4
votes
1answer
77 views

Why generalize the derivative for multivariable functions? [duplicate]

Sorry if this is a dupe (did a search, couldn't find anything). In single variable calculus, if the following limit exists: $$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},$$ then this expression ...
1
vote
2answers
34 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
0
votes
1answer
61 views

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 ...
5
votes
2answers
61 views

If $f'(x)\cdot x$ goes to zero then $f(2x)-f(x)$ is bounded.

Let $g:\mathbb R^m\to\mathbb R^n$ be defined by $g(x)=f(2x)-f(x)$ where $f:\mathbb{R}^m\to\mathbb{R}^n$ is a given differentiable function. The problem is to prove that if $\lim_{|x|\to\infty} ...
2
votes
1answer
32 views

What is a closed form expression for the ∂/∂w(∂t/∂w) if w(t) is complicated function?

Lets say we have a trigonometric function w(t) that can not be inverted as t(w). The derivative ∂t/∂w can be calculated as 1/(∂w(t)/∂t). What is a closed form expression for the second derivative ...
5
votes
5answers
76 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
0
votes
1answer
23 views

Evaluate derivatives y'(0),z'(0),y''(0),z''(0) of implicit functions y(x) and z(x)

Evaluate derivatives $y'(0)$, $z'(0)$, $y''(0)$ ,$z''(0)$ of implicit functions $y(x)$ and $z(x)$, where $y(0)=-1$ and $z(0)=1$, given by system of equations: $x+y+z=0$ and $x^2+y^2+z^2=0$ First ...
1
vote
1answer
35 views

Sufficient conditions for differentiability of multivariate functions

Claim: If a function $f:\mathbb R^2\to\mathbb R$ has partial derivatives in a neighborhood $D$ of $(x_0,y_0)$, and if these are continuous at $(x_0,y_0)$, then $f$ is differentiable at $(x_0,y_0)$ ...
2
votes
1answer
67 views

Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
0
votes
2answers
42 views

$f : \mathbb R^n \to \mathbb R$, what is the gradient of $f(tx)$?

Fairly simple question, suppose there is a function $f: \mathbb R^n \to \mathbb R$, and a scalar $t \in \mathbb R$. is it possible to find $D_f(tx)$ using only $t$ and $D_f(x)$? Perhaps using chain ...
1
vote
1answer
20 views

Optimization of parallelepiped.

Let $K \in R^3$ the ellipsoid given by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $ with $a,b,c > 0$ , let $(x,y,z) \in K$ on the first octant, consider the ...
0
votes
0answers
18 views

Prove or disprove the statement related to the definition of multivariable-differentiable function

The question: Let $f,f_1,...,f_n \; (n > 0)$ be functions from $\mathrm{D} \subset\mathbb{R}^n$ to $\mathbb{R}$ satisfying $$\left ( \sqrt{\sum_{i=1}^n x_i^2} \right ) f(\mathrm{x}) = \sum_{i=1}^n ...
0
votes
1answer
21 views

Direction for greatest derivative

Suppose I have a function like $f(x,y) = e^x e^y x^2 y^2$, and I want to know in which direction the derivative will grow fastest at a stationary point. $(0,0)$ is a stationary point of the example ...
1
vote
0answers
22 views

What is the 1st derivative i.r.t. coordinates for a vector function?

For a vector function $f(x,y,z)$, we have the divergence $$\nabla \cdot f(x,y,z) = \frac{\partial{f}_{x}}{\partial x}+\frac{\partial{f}_{y}}{\partial y}+\frac{\partial{f}_{z}}{\partial z}$$ , the ...
1
vote
1answer
40 views

Find the partial derivatives of second order of $f(x,y)=\varphi(xy,\frac{x}{y})$

Ok guys, I'm given this smooth function $\varphi(u,v)$ defined in $R^2$. So that $f(x,y)=\varphi(xy,\frac{x}{y})$. I have to find all partial derivatives of second order of $f$ using the partial ...
3
votes
1answer
53 views

Prove that a function is differentiable if…

I'm trying to prove that given a differentiable function $f: \mathbb{R}^2 \to \mathbb{R}^m$ in $p =(p_1, p_2) \in \mathbb{R}^2$, the function $$ g(x, y) = f(x, y) - \frac{\partial f}{\partial x}(p)(x ...
0
votes
0answers
20 views

What's the jacobian of this function

We are given a function $f$ that maps the coefficients of a polynomial to its roots. meaning $f(a_1,a_2,a_3)=(x_1,x_2,x_3)$ if ...
2
votes
1answer
24 views

Acquiring $Df(\mathbf{x})$

Sorry for the probably easy and silly question, but I try to teach myself linear algebra and I am stucked at "the derivative as a matrix" part. I know how to differentiate partially and I know how ...
0
votes
0answers
20 views

Derivative of bilinar map $\mathbb R^2 \times \mathbb R^2$ to $\mathbb R$

Let $f :\mathbb R^2 \times \mathbb R^2 \mapsto \mathbb R$. Then for $(V,W) \in \mathbb R^2 \times \mathbb R^2$, the derivative $D f(V,W)$ evaluated on $(H,K) \in \mathbb R^2 \times \mathbb R^2$ is ...
3
votes
0answers
36 views

Prove Differentiation Multivariable

Given $f(x,y) = \frac{ xy^2}{x^2 +y^2}$ From defintion we know it is differentiable if: $\lim_{h\to 0}\frac{F(X+h)-F(X)-c*h}{|h|}$ exists, where $c$ is the gradient of the function. I have ...
2
votes
1answer
64 views

minimum and maximum of $f(x,y)=\sin(x)+\sin(y)-\sin(x+y)$

we are asked to find the minimum and maximum of the function$f:A \to A$ $f(x,y)=\sin(x)+\sin(y)-\sin(x+y)$ Where $A$ is the triangle bound by $x=0$,$y=0$ and $y=-x+2\pi$ I'd like someone to review ...
3
votes
0answers
17 views

$\sqrt{X}$ where $X$ is a positive definite matrix is smooth $C^{\infty}$ [duplicate]

I'm trying to prove the following statement. Let $P_n \subset Mat_{nxn}(\mathbb R)$ be the set of all symmetric positive definite matrices with real entries of size $n$x$n$. Let $\sqrt{}:P_n \to ...
1
vote
2answers
66 views

Local max/min of this

$$f(x,y)=(x-y)^n$$ where $n \geq 1$ using this limitation $ x^2 + y^2 -1 =0$ I tried to find the derivatives $$ \begin{cases} f_x = 2λx + n(x-y)^{n-1}=0 \\ f_y = 2λy - n(x-y)^{n-1}=0\\ f_λ = x^2 + ...
2
votes
0answers
20 views

Rank of the differential

Let $f:\mathbb R^n \to \mathbb R^n$ such that $f$ maps roots of a polynomial to its coefficients. Meaning: if $(x-x_1)(x-x_2)...(x-x_n)=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_n$ then $f\begin{pmatrix} x_1 ...
2
votes
1answer
42 views

Find the differential of $f(A)=det(A^{-1}-A)$ where $A$ is invertible.

The question is if $A$ is an invertible matrix with real entries of size $n$. Is $f(A)=det(A^{-1}-A)$ differentiable? and what is the differential. I think I managed to show it's differentiable. the ...
0
votes
1answer
33 views

How do I prove continuity of partial derivative of $f(x, y) = \sqrt{x^4+y^4}$ at $(0,0)$?

Consider I have $$f(x, y) = \sqrt{x^4+y^4}$$ And I want to check if the function has partial derivatives continuous in point $$(x_0, y_0) = (0, 0)$$ I know theorem, that existence of continuous ...
2
votes
1answer
47 views

Smoothness of $f(x)/(1+|f(x)|)$ where $f \in C^1(E)$ for $E$ an open subset of $\mathbb{R}^n$

(a) Show that if $E$ is an open subset of $\mathbb{R}$ and $f \in C^1(E)$ then the function $$F(x) = \frac{f(x)}{1+|f(x)|}$$ satisfies $F \in C^1(E)$. (b) Extend the results of part (a) to $f \in ...
2
votes
1answer
33 views

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 ...
2
votes
1answer
37 views

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 ...
0
votes
1answer
51 views

Given that $|f(x,y)| \le x^{2}y^{2}$, prove that $f$ is differentiable at (0, 0).

Given that $f : \mathbb{R}^{2}\rightarrow \mathbb{R}$ is a function such that $|f(x,y)| \le x^{2}y^{2}$ for all $(x,y) ∈ R^{2}$, prove that $f$ is differentiable at $(0, 0)$. I know that I should ...
1
vote
2answers
28 views

A question on a multivariable continuously differentiable function

Assume $f(x_{1},x_{2})$ is a real-valued continuously differentiable function, and assume it holds that $x_2D_{1}f(x_1,x_2) - x_1D_2f(x_1,x_2) = 0$ where $D_1$ is the partial derivative with respect ...
2
votes
1answer
29 views

Direction of gradient from level surface?

In the diagram below, we see a level surface with a gradient. As a consequence of the multivariable chain rule, the gradient is normal to the surface. That's clear to me. Why is the gradient ...
1
vote
0answers
22 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 ...
2
votes
1answer
40 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 ...
0
votes
0answers
26 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} ...
1
vote
1answer
22 views

Is there any trick you can use to derive f( h(x),x)

I was just wondering, is there a way to derive $ \frac{d}{dx} f( h(x),x)$ without knowing how the function looks? For example by some trick of using multivariable diferentiation of $f(h(x),y)$? Thank ...
1
vote
1answer
60 views

Why can't I use the chain rule on $f(x,y) = \frac{xy^2}{x^2 + y^2}$?

The function is $f(x,y) = \frac{xy^2}{x^2 + y^2}$ and $0$ if $(0,0)$. So $f(x,y)$ is continous, the partial derivatives exist, and the partial derivatives are also continuous (the limit as $(x,y) ...
1
vote
2answers
58 views

Question about limit definition of partial derivative

I've seen it written two different ways: $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x + h, y) - f(x,y)}{h}$$ and $$\frac{\partial f}{\partial x} = \lim\limits_{h ...
0
votes
1answer
30 views

Question about definition of derivative

$$\lim\limits_{\textbf{x} \rightarrow \textbf{x}_0} \frac{\|f(\textbf{x}) - f(\textbf{x}_0) - \textbf{T}(\textbf{x} - \textbf{x}_0)\|}{\|\textbf{x} - \textbf{x}_0\|} = 0$$ Does the $\textbf{T}$ part ...
0
votes
1answer
40 views

Help with definition of derivative

My textbook says the definition is this: $$\lim\limits_{\textbf{x} \rightarrow \textbf{x}_0} \frac{\|f(\textbf{x}) - f(\textbf{x}_0) - \textbf{T}(\textbf{x} - \textbf{x}_0)\|}{\|\textbf{x} - ...
0
votes
0answers
22 views

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 ...
0
votes
1answer
30 views

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 ...
1
vote
1answer
24 views

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?