0
votes
2answers
60 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
1
vote
2answers
35 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 ...
-1
votes
1answer
25 views

derivative of sum of vectors

suppose i need to make the partial derivative of this vector function $f(\vec{a},\vec{b})=\frac{1}{| \vec{a}+\vec{b}|}$ respect to $\vec{a}$: $\frac{\partial }{\partial \vec{a}} f(\vec{a},\vec{b})$, ...
0
votes
0answers
31 views

Is it correct that $\frac{d\theta}{d\varphi'}=\left(\frac{d\varphi'}{d\theta}\right)^{-1}$?

Let $\theta$ be $k\times1$ and $\varphi$ be $k\times1$. Then $\frac{d\theta}{d\varphi'}$ is a $k\times k$ matrix with entries $\frac{d\theta_{i}}{d\varphi_{j}},\,\, i,j=1,...,k$. I was wondering if ...
1
vote
0answers
14 views

Differentiation of vector-function

Let $f(x) = e^{-x^Tx},$ where $x \in \mathbb{R}^n$. What will be the second derivative? The first is $~f'(x) = 2x^T e^{-x^Tx}$, and when I try to find the second, I confuse. It will be $$f''(x) = ...
3
votes
0answers
34 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 ...
0
votes
0answers
17 views

Derivative of a parametrized vector on a nonfixed basis

Suppose a curve defined by a vector parametrized through the variable $u$, and expressed on a non-fixed base, like the polar coordinates base. You derive it with respect to that parameter. What ...
1
vote
2answers
68 views

Multivariable Calculus - Calculating Derivative Matrix

I'm working with Munkres' Analysis on Manifolds. From chapter 2 (this isn't a homework question): Given $f: \mathbb R^2 \rightarrow \mathbb R^2 : f(r,\theta)=(r\cos(\theta),r\sin(\theta))$, ...
1
vote
3answers
80 views

Differentiating a non-linear functional with respect to a vector

I have the functional: $$F=v^T\times A \times v$$ Where $A$ is a function of $v$. The non-linear system of equations necessary to find $v$ is obtained doing: $$\frac{\partial F}{\partial v}=0$$ ...
0
votes
1answer
273 views

Divergence of this field, spherical coordinates

Given this field $$ \mathbf{F(\mathbf{r})}=F_0a^3\left( \frac{2\cos\theta}{r^3}\mathbf{\hat{r}}+\frac{\sin\theta}{r^3}\mathbf{\hat{\theta}}\right) $$ How can I show that the divergence is zero ($r\neq ...
3
votes
2answers
161 views

Second derivative is what?

I wonder what is the meaning of the second derivative or what kind of object it is when we have a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$. The first derivative is the Jacobian matrix, but ...
0
votes
1answer
56 views

Derivative of a vector function

Can someone please check my work below to confirm whether or not I got the correct answer? This is question 13.2.16 in the 7th edition of Stewart Calculus. Find the derivative of the vector function: ...
1
vote
2answers
324 views

How do I calculate numerically a tensor in polar coordinates?

You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid? a) fit a (spline) surface, ...
3
votes
1answer
110 views

Is this vector derivative correct?

I want to comprehend the derivative of the cost function in linear regression involving Ridge regularization, the equation is: $$L^{\text{Ridge}}(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T\beta)^2 + ...
1
vote
1answer
174 views

Gradient vector function using sum and scalar

Could someone take a look on my attempt to compute the gradient for: $$f(x) = \lambda \sum_{x = 1}^n g(x_i)$$ Where $x \in \mathbb{R^d}$, $\lambda \in \mathbb{R}$ and $$g(x_i) = \begin{cases} x_i - ...
2
votes
1answer
614 views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
1
vote
1answer
55 views

Vector derivative with power of two in it

I want to compute the gradient of the following function with respect to $\beta$ $$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2$$ Where $\beta$, $y_i$ and $x_i$ are vectors. The ...
1
vote
2answers
747 views

Log-likelihood gradient and Hessian

Considering a binary classification problem with data $D = \{(x_i,y_i)\}_{i=1}^n$, $x_i \in \mathbb{R}^d$ and $y_i \in \{0,1\}$. Given the following definitions: $f(x) = x^T \beta$ $p(x) = ...
1
vote
1answer
95 views

Vector derivatives, what is the minimum of this matrix equation?

I am new to vector derivatives and trying to figure out a lot for my Machine Learning course. I have given the following: $x \in \mathbb{R}^n$, $y \in \mathbb{R}^d$, $A \in \mathbb{R}^{d \times n}$, ...
3
votes
2answers
287 views

Need help computing the partial derivatives of a vector function.

I need help computing the partial derivative shown below. I've never taken a course in vector analysis so I'm not if my previous attempts at solving the problem were even on the right track. If ...
3
votes
2answers
287 views

Differentiating a function with respect to a vector

I need to differentiate the function $u$ shown below with respect to a vector $\psi$: ($a, c$ and $f$ are constants) $$u(\psi) =\left[\begin{array}{cccc} a & f & 0 & 0\\ c & a & f ...