Intro to Tensors
Vectors?
In order to understand what a tensor is, it is helpful to understand what a vector is. A vector is a mathematical object which has both magnitude and direction. The magnitude of a vector is a scalar quantity which describes the vector’s length. To define the direction of a vector, we must specify a basis. For example a basis for \(\mathbb{R}^2\) (the plane in two dimensions), is given by the unit vectors \(\vec{e}_1\) and \(\vec{e}_2\), where
\(\vec{e}_1 =
\begin{pmatrix}
1\\
0
\end{pmatrix}
\text{ and }
\vec{e}_2 =
\begin{pmatrix}
0\\
1
\end{pmatrix}\)
These two vectors define (in the plane) the horizontal (\(\vec{e}_1\)) and vertical direction (\(\vec{e}_2\)). So every point on the plane is a combination of these two vectors. For example, consider the vector \(v\) given by
So the direction of \(v\) in this basis is \(3\) to the left and \(4\) units up. Notice that our basis depends on our coordinate system. In the exmaple above, we chose cartesian coordinates, but we could just as well chose polar coordinates, or some other coordinate system. If we had done so, we might have to find a different basis. This just means we are using different “directions” to describe the same object. We could say that an object is positioned 3 units to the left and 4 units upward from some specified origin, or we could say that it is 5 units long and at an angle 127 degrees from the horizontal axis. Both descriptions provide us with the same vector, but use different directional rules (basis vectors) describe the same vector. What changes are the components of the vector.
The components in the new basis can be determined by using a transformation of the old components. This is generally a matrix whose entries describe how each component changes with respect to the coordinate in the other;