Hi there, I have edited the initial post, apologies, I had put the unit vectors j and k in the wrong place, thankyou for correcting me. I have also included an extra line showing where the acceleration vector comes from, it is simply the second derivative of position with respect to phi + the...
Homework Statement
The following question involves a torque acting on a particle in rotational motion. It provides practice with the various equations for angular velocity, torque etc A particle of mass ##m## initially has position...
Ok, let me just get a few things cleared up in my understanding..
Firstly, the questions I am attempting to solve are on Einstein summation convention, so the question should be written..
$$\sum_{j=1}^{3} x_{j}b_{ij} = x^{j}b^{i}_{j}$$
Which means that the sum is equal to ...
Ok, I think I understand, many thanks for your patient help.
Just to make sure I've understood correctly, if we take another example:
$$\sum_{j=1}^{3} x_{j}b_{ij}$$
This would then equal..
$$x_{1}b_{i1}+x_{2}b_{i2}+x_{3}b_{i3}$$
Which is the dot product of the matrices...
so, just to make sure I am with you so far, the entry position of ##(A \cdot x)_{i}##
$$=\sum_{j=1}^{n} a_{ij}x_{j}$$
Where ##j## represents columns in the matrix ##A## and rows in the matrix ##x##
and
$$(A \cdot x)=Ax$$
where A is a square matrix ...
$$(AB)_{ik}=\begin{pmatrix}a_{11}b_{11}+a_{12}b_{21}&a_{11}b_{12}+a_{12}b_{22}\\a_{21}b_{11}+a_{22}b_{21}&a_{21}b_{12}+a_{22}b_{22}\end{pmatrix}$$
and
$$(A(\cdot)x)_{i}=a_{i1}x_{1}+a_{i2}x_{2}+... a_{in}x_{n}$$
May I ask what the correct latex code for A dot x is?
For the first task.
$$
\begin{bmatrix}a_{ij}&a_{ij}&a_{ij}\end{bmatrix}\cdot\begin{bmatrix}b_{jk}\\b_{jk}\\b_{jk}\end{bmatrix}$$
And for the second tast...
To write the original question in Einstein summation, it would be.. ?
$$a_{ij}b_{jk}x_{k}$$
for (i,k) = (2,3) we get.. ?
$$a_{2j}b_{j3} = a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}$$
And in the ith row and kth column we would have... ?
$$a_{i1}b_{1k}+a_{i2}b_{2k}+a_{i3}b_{3k}$$
So, the inner sum has one repeated indicie and two free indices, so therefore has nine equations, forming a 3x3 matrix with three terms in each equation, giving us...
$$...
Homework Statement
The question asks us to write down the matrix represented by the following summation.
2. Homework Equations
The question summation...
$$\sum_{j,k=1}^{3} a_{ij}b_{jk}x_{k}$$
The Attempt at a Solution
$$
\sum_{j,k=1}^{3} a_{ij}b_{jk}x_{k} =...
Homework Statement
Hi there, I am attempting to prove a trigonometric equation using the half angle and double angle formulae
Homework Equations
See image one...
The Attempt at a Solution
See image two...
I get stuck after the second line and can't see how to continue, please help.[/B]