Tensor Discussion One

Tensor Calculus is constructed so that it can do something helpful to a physics story without changing the physics story.

If a bull is charging towards you and it will get to you in 5 seconds it doesn’t matter if we rotate the coordinates and change the units from inches to centimeters, that bull will still get to you in five seconds.

There is only one scenario where a transformation acting on a letter with an index moves the index and we keep the letter the

Appendix A

$a_{ij} b_{jk} = c_{ik}$ for multiplication of a matrix by a matrix
$a_{ij} \: b_j = c_i$ for multiplying a matrix against a vector

For both stories above, the algebraic object a is a tool that changes b to c.

That a is not a tensor.

Appendix W

Summations commute:

$\displaystyle \sum_{a=1}^2 \sum_{b=1}^2 = a_1(b_1 + b_2) + a_2(b_1 + b_2)$

$\displaystyle \sum_{b=1}^2 \sum_{a=1}^2 = b_1(a_1 + a_2) + b_2(a_1 + a_2)$

If we expand out both answers we will see that each contains the following four terms:

$a_1 b_1$
$a_1 b_2$
$a_2 b_1$
$a_2 b_2$

Definitions/Protocols:

Vector v is overed by an arrow
Covector w is overed by a tilda

Contravariant indices go superscript
Covariant indices go subscript.

One can regard an electric field as a gradient, 5.5 on page 27

Derivatives of tensors do not transform as tensors.

We are interested in vectors and one-forms as geometric objects, not as list of components tied to some arbitrary coordinate system.

If we choose to work in an orthonormal system then we will make all indices subscripts so that it doesn’t look like we’re distinguishing between covariant and contravariant vectors.

A Kronecker Delta can rename an index.

If tensors are of the same type, one can add them. One should then choose the indices to be the same.

The product of tensors is a tensor if in each summation the summation takes place over one upper index and one lower index.

Second-rank quantities such as stress, strain, moment of inertia, and curvature can be denoted as 3×3 matrix arrays;

The area of a triangle bounded on two sides by vectors a and b is (1/2) a x b ;

It is half the cross product of A and B

Appendix X

$\overrightarrow{e_i}' = \sum_j a_{ij} \: \overrightarrow{e_j}$

For the above, the prime symbol indicates a new basic vector and the lack of a prime indicates an old basis vector.

Component $a_{ij}$ calculates how much of $\overrightarrow{e_j}$ is in $\overrightarrow{e_i}'$ .

We can show this with a simple example:

$\begin{bmatrix} 6 \\ 6 \end{bmatrix} = \begin{bmatrix} 2 && 0 \\ 0 && 2 \end{bmatrix} \begin{bmatrix} 3 \\ 3 \end{bmatrix}$

How much of 3 is in 6? We have 2 three’s in six. Note, it’s very simple for this example. In another problem we could have two or more old components contributing to the new component.

Personally lwe would have preferred to write, how many times does 3 occur in 6?

Appendix Y

This is sometimes called component-wise addition.

For tensor addition, add corresponding elements.

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