0
votes
1answer
18 views

Tangent plane that passes through a point

How would I find the (a,b) that satisfies that the tangent plane to $f(x,y) = (x^2) + 2xy + (y^2)$ passes through the point $(2,1,0)$ ? I know that $f(x)= 2x + 2y$, and $F(y): 2x + 2y$. Therefore ...
0
votes
0answers
11 views

Partial derivative of a Piecewise function with two variables

I'm having trouble solving partial derivatives of a piecewise function where the function breaks. For the following equation, what would the partial derivatives (both x and y) be? f(x,y): {x if ...
1
vote
1answer
14 views

Tangent planes perpendicular at each point of intersection

Find the set of all points $(a,b,c)$ in 3-space for which the two spheres $(x-a)^2+(y-b)^2+(z-c)^2=1$ and $x^2+y^2+z^2=1$ intersect orthogonally.( Their tangent planes should be perpendicular at each ...
2
votes
3answers
66 views

Lagrange multipliers from hell

I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way: "Find the closest and furthest points on the circle made from the intersection of the ...
2
votes
0answers
21 views

Total derivative

What is the significance and meaning of the total derivative? Why is it introduced in the definition of differentiability of scalar and vector fields?
1
vote
2answers
18 views

Application of chain rule

The equations $u=f(x,y),x=X(t),y=Y(t)$ define $u$ as a function of $t$, say $u=F(t)$. Compute $F'(t)$ in terms of $t$ if, $$f(x,y)=\log [(1+e^{x^2})/(1+e^{y^2})] , X(t)=e , Y(t)^t=e^{-t}.$$ From the ...
0
votes
0answers
14 views

Multivariable differentiation under the integral sign

Suppose that the functions $f:[a,b]$x$[a,b] \to \mathbb{R}$ and $\frac{\partial{f}}{\partial{t}}:[a,b]$x$[a,b] \to \mathbb{R}$ are continuous. Prove that the function $F:[a,b]$x$[a,b] \to \mathbb{R}$ ...
0
votes
0answers
16 views

Nonexistence of a scalar field

Prove that there is no scalar field $f$ such that $f'(a;y)>0$ for a fixed vector $a$ and every non-zero vector $y$.
1
vote
1answer
25 views

Constant function on a convex set

If $f'(x;y)=0$ for every $x$ in an open convex set $S$ and every $y$ in $R^n$, prove that $f$ is constant on $S$. A set $S$ is called convex if for every $a$ and $b$ in $S$, ${ta+(1-t)b \epsilon S}.$ ...
1
vote
0answers
46 views

What exactly do the terms of $(f \circ g)'''$ mean?

Say, $g: X\to Y$ and $f: Y\to Z$ are smooth. One can find $(f \circ g)'''$ by using the FaĆ  di Bruno's formula: $$(f \circ g)''' =(f'''\circ g)(g')^3 + 3(f''\circ g)g'g'' + (f'\circ g)g'''$$ But my ...
1
vote
0answers
18 views

Hessian matrix of $g\circ f$

Say, $f:\mathbb R^n\to\mathbb R^k$ and $g:\mathbb R^k\to\mathbb R$ are both $C^2$. I'd like to express the Hessian matrix of $g\circ f$ $$\left( \frac{\partial^2(g\circ f)}{\partial x_i \partial ...
0
votes
2answers
20 views

Maximize the directional derivative

Find the points $(x,y)$ and the directions for which the directional derivative of $f(x,y)=3x^2+y^2$ has its largest value, if $(x,y)$ is restricted to be on the circle $x^2+y^2=1$. For the point ...
1
vote
1answer
40 views

Directional Derivative and differentiability

My question is similar, but not equal to this...Question on linearity of directional derivative Let $f'_{h}(a)$ be the directional derivative. And for the function $f:\mathbb{R}^n\rightarrow ...
0
votes
2answers
44 views

Higher Order Partial Derivatives

If i have 3 times differential function $ z= f(x^3 / y^4) $ how can i get: a) ${\partial z \over \partial x}$ b) ${ \partial ^2z \over \partial x^2}$ c) ${\partial^2z \over \partial x \partial ...
0
votes
0answers
14 views

Second order of total differential of function:

If i have function $ z(x, y)= x^2 + e^{x*y} -y^3 $ how can i find 2nd order total differential? Can someone explain me step by step please.
0
votes
2answers
27 views

Help understanding question regarding 3rd derivative and “smallest uniform bound”?

I'm a big user of Stack Overflow, however, a first time user here. I'm working on a problem for a math class that's pretty easy (I'm sure), I just don't understand the question really. Here it is ...
1
vote
2answers
39 views

Is it possible to find a function if we know its differential?

Not something we were taught at uni yet, just something that peaked my curiosity. If I was given a derivative of a scalar function, for example $f'(x)=x$ then I know that $f(x)=\frac{x^2}{2}$ (let's ...
0
votes
0answers
18 views

Hessian of a non-linear Matrix function

Apologies if this is a silly question, but I am really confused. I am trying to find the Hessian of a non-linear function $f$. I understand that the Hessian of $f$ with respect to $A$ is the Jacobian ...
1
vote
1answer
22 views

Problem about partial derivatives

I have two problems: Let $$f : \mathbb{R}^2 \to \mathbb{R}, f \in C^2$$. Problem 1 Find all functions such that $$\frac{\partial^2f}{\partial x \partial y} = 0$$ Problem 2 Find all functions ...
0
votes
1answer
19 views

Continuity of a partial derivative

I have the function $$f(x,y)=\begin{cases} x^2ysin(\frac1x) & \text{if $x$ is not 0} \\ 0 & \text{if $x=0$}\end{cases}$$ And I need to find the derivative and the ...
2
votes
1answer
47 views

Very basic questions on chain rules and product rules

I have serious gaps in maths and would like to ask some basic questions. I know there is the following chain rule for the first derivative: $$ Dh(x) = Dg(f(x))Df(x)\quad\quad (1) $$ where $h(x) = ...
2
votes
0answers
24 views

A vector analysis question requiring multivariable calculus

The question is taken from a Vector-Analysis worksheet as an extension exercise, here is the first part: Let $\underline{v}$ be a vector field $\mathbb{R}^{2} \backslash (0,0)$ of the form ...
2
votes
3answers
57 views

Differentiation of functions w.r.t. a composed argument

I need help with the following derivative involving inner products: $$\frac{d\, \log(x)^T\,y}{d\,x^T\,y}$$ Here $x$ and $y$ are $n$-dimensional vectors, $T$ indicates transpose, and the logarithm of ...
1
vote
1answer
30 views

Help with multivar. chain rule

I am having trouble with the following problem. I feel that I do understand the multivariable chain rule in general, but applying it here is more difficult. I am lost on where to start. Any help would ...
-1
votes
2answers
21 views

Writing out chain rule for the following function

$\frac{dh}{dx}$, where $h(x) = f(x, u(x), v(x))$. First of all, this function doesn't even make sense to me. It's a function of one variable, with domain $\mathbb{R}$ and range $\mathbb{R}$. How can ...
0
votes
2answers
25 views

Is there an easier way to prove a multivariate function is differentiable?

$f\colon U \rightarrow \mathbb{R}, (x,y) \mapsto \sqrt{1 - x^2 - y^2}$ where $U = \{(x,y) \mid x^2 + y^2 < 1\}$. So the definition of differentiability I have is: $$\lim \limits_{(x,y) ...
1
vote
1answer
66 views

Show a function is not continuous at a point

$$ f(x,y) = \begin{cases} \dfrac{x^2 y^4}{x^4 + 6y^8}, & \text{if }(x,y)\neq(0,0) \\ 0, & \text{if }(x,y)=(0,0) \end{cases} $$ For the definition of differentiability, I have: $$\lim_{h ...
0
votes
0answers
36 views

Totally differentiable function - definition

I know for a function of several variables, if all partial derivatives exist and they are continuous at and around a point $a$ then the function is totally differentiable at that point. I ...
1
vote
1answer
60 views

Very interesting multivariable calculus question.

If $\displaystyle z = \frac{f(x-y)}{y}$, show that $\displaystyle z + y \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 0$.
1
vote
1answer
54 views

Inverse function theorem question - multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) ...
0
votes
0answers
13 views

How do you know when the derivative is a matrix or a sum?

For the derivative of a function, sometimes I see the derivative written as a matrix of the partial derivatives, and sometimes I see the derivative written as a sum of the partial derivatives. When do ...
0
votes
1answer
49 views

Trouble reading directional derivative proof

I'm reading Vector Calculus from http://mecmath.net/. This is a free PDF book for students of Calculus III. In section 2.4 (page 78) it introduces the directional derivative and theorem 2.2: So far ...
1
vote
2answers
66 views

Multivariable calculus - explain what the teacher did

The teacher gave this exercise: Find $D_f(a)$ when $f: \mathbb R^n \to \mathbb R$, $f(x)=<x,\xi>^2$ where $\xi \in \mathbb R^n$. What I did: I wrote it as $$f(x)= (\sum_{i=1}^{n}x_i ...
0
votes
0answers
46 views

Taylor series like polynomials

Let $U$ be an open subset of $R^n$ and $f:U\rightarrow \mathbb{R}$ a function and $x\in U$ such that in a small neighbourhood of $x$ and for $\epsilon \in \mathbb{R^b}$ sufficiently small we have the ...
-1
votes
1answer
40 views

continuity and differentiability of function of two variables

Let $f(x,y)$ be $$f(x,y): \begin{cases} x & \text{for } y = 0\\ x-y^3\sin\left(\frac{1}{y}\right)& \text{for } y \neq 0\end{cases} $$ then check continuity and differentiability at $(0,0)$. ...
0
votes
1answer
62 views

Check my answer - Differential of $P(A)=\det(A^{-1}-A)$

We are asked to find the differential of $P: GL_n(\mathbb R) \to \mathbb R$, $P(A)=\det(A^{-1}-A)$ and show it is differentiable. If we define $f(A)=\det(A)$ and $g(A)=A^{-1}-A$ then it is clear ...
0
votes
1answer
22 views

Finding directional derivatives that exist

Let $$g(x,y,z)= \begin{cases} \frac{xy+xz+yz}{\sqrt{x^2+y^2+z^2}}, & \text{if } xi+yj+zk \neq 0 \\ 0, & \text{if } xi+yj+zk = 0 \\ \end{cases} $$ Use the definition of the directional ...
3
votes
1answer
35 views

Prove that a function is differentiable using the limit definition

Use the definition of the derivative to prove that $f(x,y)=xy$ is differentiable. So we have: $$\lim_{h \to 0} \frac{||f(x_0 + h) - f(x_0) - J(h)||}{||h||} = 0$$ We find the partial derivatives which ...
1
vote
0answers
44 views

check my answer - Show that $f(A)=trace(A^2)$ is differentiable and find the differential at any point

As topic says, we are given $f: Mat_n(\mathbb R) \to \mathbb R,f(A)=trace(A^2)$ where $A$ is an n by n matrix with real entries. I think I managed to show that $f$ is both differentiable, and find ...
1
vote
1answer
33 views

differential (Jacobi Matrix) of $f(A)=A^2$ where $A$ is a matrix - check my answer

I just want a quick verification that what I did here is correct: let $f(A)=A^2$ where $A$ is a n by n matrix with real entries. then $$D_f(A)=\lim_{t \to 0} \frac{f(A+tA)-f(A)}{t} = \lim_{t \to 0} ...
1
vote
1answer
44 views

Infinitesimals in gradients

Take the function $y(\vec v)$ such that $y:\mathbb R^n\to\mathbb R$. Given it's gradient $\nabla y = \left(\frac{\partial y}{\partial v_1},\cdots,\frac{\partial y}{\partial v_n}\right)$, it is ...
1
vote
1answer
39 views

Check my answer - Finding the jacobi matrix of a function

We are given $f: \mathbb R^n \to \mathbb R^n$ such that: $0 \neq x \in \mathbb R^n$, $f(x)=\frac{x}{|x|}$, where $|x| = \sqrt {x_1^2 +x_2^2+...+x_n^2}$ Find the jacobi matrix (the differential ...
0
votes
1answer
28 views

introductory calculus - Help me find a function with a few properties

I was asked to find a function $f: \mathbb R^2 \to \mathbb R$ such that: 1) $f$ is continuous at $(0,0)$. 2) $f$ has directional derivatives at $(0,0)$ (does this mean $f$ is differentiable at ...
1
vote
1answer
28 views

Question on Gradients

Consider the following vector function $y: \mathbb R^n \to \mathbb R$ $$ y(\vec x) = y(x_1,x_2,\cdots,x_n)$$ Is it correct to state the following? $$ dy = \sum_{i=1}^{n}{\left(\frac{\partial ...
1
vote
1answer
127 views

Derivative of sum of two functions is the sum of their derivatives.

Suppose $x_0 \in U \subseteq \mathbb{R}^d$, $U$ open, and $f,g : U \to \mathbb{R}^m$ differentiable at $x_0$, then $$D_{f + g} (x_0) = D_f(x_0) + D_g(x_0).$$ MY ATTEMPT Put $r(x) = f(x) + g(x) ...
3
votes
1answer
54 views

Differentiable function in n dimensions

If a function $f:\mathbb R^{n} \rightarrow \mathbb R^m $ is differentiable at a point $a$ can we say that there is a neighbourhood of $a$ such that $f$ is locally Lipschitz? (i.e. there is some ...
1
vote
1answer
46 views

Partial derivative on convex set

If we have a function $f:U \rightarrow R$ ($U \subset R^n$) which is partially differentiable on a convex set U with $\frac{\delta f}{\delta x_1} = 0$ for all $x \in U$. How can we prove that $f$ ...
2
votes
1answer
60 views

Differentiability in $R^n$

I have the definition of the derivative for $f:\mathbb R^n \rightarrow\mathbb R^m$ at a point $a$ as: $f$ is differentiable at a then there exists a linear map $L:\mathbb R^n \rightarrow\mathbb ...
1
vote
1answer
25 views

Partial differentiation of a composite function

This should be straightforward, but don't seem to be able to crack it. Take a function $f(x_1, x_2, x_3)$ and a function $g(x4, x5, x6)$. These two functions mapp from $R^3 \rightarrow R^1$. I am ...
0
votes
2answers
144 views

Question about the differential

Today at class, my teacher stated the following proposition saying it is obvious: Let $x_0 \in U \subset \mathbb{R}^d$, $U$ open, and $f: U \to \mathbb{R}^m$ differentiable at $x_0$, then for any $v ...