Let us imagine that we have a system of coordinates and a system of coordinates that is rotated relatively to . Let us consider a point that has coordinates on and coordinates on .
In general it is obvious that , and that .
Since the transformation from to is just a rotation we can assume that the transformation is linear. Hence we can write explicitly
Another way to write the three previous equations in a more compact way is:
In case you don’t see how the previous equation is a more compact way of writing the first equations I’ll just lay out the case.
Now all that we have to do is to sum from to and we get the first equation. For the other two a similar reasoning applies.
If we want to make a transformation from to the inverse transformation is
The previous notation suggests that the indexes can be arranged in a form of a matrix:
In the literature the previous matrix has the name of rotation matrix or transformation matrix.
— 1. Properties of the rotation matrix —
For the transformation
Where is a matrix known as Kronecker delta and its definition is
For the inverse transformation it is
The previous relationships are called orthogonality relationships.
— 2. Matrix operations, definitions and properties —
Let us represent the coordinates of a point by a column vector
Using the usual notation of linear algebra we can write the transformation equations as
Where we define the matrix product, , to be possible only when the number of columns of is equal to the number of rows of
The way to calculate a specific element of the matrix , we will denote this element by the symbol is,
Given the definition of a matrix product it should be clear that in general one has
As an example let us look into;
We’ll say that is the transposed of and calculate the matrix elements of the transposed matrix by . In a more pedestrian way one can say that in order to obtain the transpose of a given matrix one needs only to exchange its rows and columns.
For a given matrix it exists another matrix such as . The matrix is said to be the unit matrix and usually one can represent it by .
If , then and are said to be the inverse of each other and , .
Now for the rotation matrices it is
Where the second last equality follows from what we’ve seen in section 1.
Just to finish up this section let me just mention that even though, in general, matrix multiplication isn’t commutative it still is associative. Thus . Also matrix addition has just the definition one would expect. Namely .
If one inverts all three axes at the same time the matrix that we get is the so called inversion matrix and it is
Since it can be shown that rotation matrices always have their determinant equal to and that the inversion matrix has a determinant we know that there isn’t any continuous transformation that maps a rotation into an inversion.
— 3. Vectors and Scalars —
In Physics quantities are either scalars or vectors (they can also be tensors but since they aren’t needed right away I’ll just pretend that they don’t exist for the time being). These two entities are defined according to their transformation properties.
Let be a coordinate transformation, , if it is:
- then is said to be a scalar.
- for , and then is said to be a vector.
— 3.1. Operations between scalars and vectors —
I think that most people in here already know this but in the interest of a modicum of self containment I’ll just enumerate some properties of scalars and vectors.
- is a vector.
- is a scalar.
As an example we will show the second proposition 5 and the reader has to show the veracity of the last proposition.
In order to show that is a vector we have to show that it transforms like a vector.
Hence transforms like a vector.
— 4. Vector “products” —
The operations between scalars are pretty much well know by everybody, hence we won’t take a look at them, but maybe it is best for us to take a look at two operations between vectors that are crucial for our future development.
— 4.1. Scalar product —
We can construct a scalar by using two vectors. This scalar is a measure of the projection of one vector into the other. Its definition is
For this operation deserve its name, one still has to prove that the result indeed is a scalar.
First one writes and , where one changes the index of the second summation because we’ll have to multiply the two quantities and that way the final result can be achieved much more easily.
Now it is
Hence is a scalar.
— 4.2. Vector product —
First we have to introduce the permutation symbol . Its definition is if two or three of its indices are equal; if is an even permutation of (the even permutations are , and ); if is an odd permutation of (the odd permutations , and ).
The vector product, , of two vectors and is denoted by .
To calculate the components the components of the vector the following equation is to be used:
Where is shorthand notation for .
As an example let us look into
where we have used the definition of throughout the reasoning.
One can also see that (this another exercise for the reader) and that .
If one only wants to know the magnitude of the following equation should be used .
After choosing the three axes that define our frame of reference one can choose as the basis of this space a set of three linearly independent vectors that have unit norm. These vectors are called unit vectors.
If we denote these vectors by any vector can be written as . We also have that and . Another way to write the last equation is .
— 5. Vector differentiation with respect to a scalar —
Let be a scalar function of : . Since both and are scalars we know that their transformation equations are and . Hence it also is and
Thus it follows that for differentiation it is .
In order to define the derivative of a vector with respect to a scalar we will follow an analogous road.
We already know that it is hence
where the last equality follows from the fact that is a scalar.
From what we saw we can write
Hence transforms like the coordinates of a vector which is the same as saying that is a vector.
The rules for differentiating vectors are:
The proof of these rules isn’t needed in order for us to develop any kind of special skills but if the reader isn’t very used to this, then it is better for him to do them just to see how things happen.