Part 4B : Tensors, Scalars, Vectors, and Matrices

Tensor

An tensor is an array of data (numbers, functions, etc.) which is expanded in any number (0 and greater) of dimensions. The number of dimensions is called rank of tensor.

Rank 0 tensor

A tensor that has no dimensions (0).

A is a 0 dimensional tensor

Rank 1 tensor

A tensor that is expanded in only one dimension.

Examples of 1 dimensional tensors

Rank 2 tensor

A two dimensional tensor

Rank 3 tensor

A 3 dimensional tensor is like matrices places one after another

As shown in figure, a rank 3 tensor has a cube (or cuboid like structure).

If rank of a tensor exceeds 3, it becomes difficult to visualize.

Dan Fleisch has given an amazing explanation of tensors

Scalar

A scalar is 0 rank tensor. In physics, various quantities are represented as a scalar such as : Distance (500 km), Temperature (10ºC), Speed (34 km/h), etc.

Vector

A rank 1 tensor is called a vector. Physical quantities such as Velocity (10 m/s), Displacement (54 m towards East), Electromagnetic Field (1 V/m).

Difference between scalar and vector :

A quantity that does not require additional information (such as direction) with it (like, temperature) is represented as scalar. Whereas, a quantity that needs direction to be specified alongside its magnitude is represented with a vector (like, electric field).

E is a vector or a rank 1 tensor

A vector is denoted with bold letter (like “E” ) or an arrow above the letter.

To plot a vector we use its elements as the value of coordinates (x, y and z axis respectively). Here, first element (0.5) is taken as x value and second element (also, 0.5) is taken as y value (if we had three elements, the third one would’ve been z value).

Vector E plotted on graph as a blue dot

After plotting the vector as a point we put an arrow to it, from origin(0,0).

A vector is simply a matrix that has either one row (called column vector) or one column (called row vector).

Matrix

A matrix is a rank 2 tensor. We have looked at matrix earlier.

A tensor with rank greater than 2 is simply called “tensor” (there is no specific name for tensors with rank more than 2).

Concept of tensor generalizes the matrices, vectors and scalars under one roof (they all are tensors but with different ranks).

Matrices as product of vectors :

When two vectors are multiplied they form a matrix.

Vector X (of order 3×1) is going to be multiplied with Vector Y (of order 1×3)

Result is a matrix Z (of order 3×3)

Vector X and Y combined, have 6 elements but their product alone has 9 elements. Thus, some matrices could be broken down as a product of two vectors.

Additional Resource :

An Introduction to Tensors for Students of Physics and Engineering by Joseph C. Kolecki

Read Part 5 : Row Picture and Column Picture

You can view the complete series here
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