Asked by: Matthew Allen

A scalar quantity is simply a number -- it has only magnitude. A scalar can be designated a tensor of rank zero.

A vector quantity has magnitude and direction. In two dimensional space, for example, it was x- and y-components, and in three dimensional space, it has 3 components. Vectors can have any number of dimensions. These components are commonly shown in a one dimensional column matrix.

a b v = c . . nA vector can be designated a tensor of rank one.

A tensor of rank two is represented by a matrix:

aa ab ac ... an T2 = ba bb bc ... bn ca cb cc ... cn . . . . . . . . ma mb mc ... mnA rank-three tensor is represented with a cubic matrix, with components coming out of your computer screen.

(Tensors with rank higher than three are harder to represent; the most common notation is known as Einsteinian Notation, which makes use of indices. Note that a rank-four tensor is represented by a hyper-rectangular matrix. )

Visualizing tensors is very difficult, akin to visualizing hyperdimensional objects. One way to think of tensors is in terms of fields.

A scalar field is created by simply assigning scalar quantities (numbers) to each point in space. Think of temperature -- each point in the room has a different temperature.

A vector field is created by assigning vectors to each point. An electric field is an example -- a test charge placed at a point in space will move at a certain speed and direction as represented by the vector at that point.

A tensor field has a tensor corresponding to each point space. An example is the stress on a material, such as a construction beam in a bridge.

Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor.

Answered by: Aman Ahuja, Physics Student, WPI, Massachussets

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(*1918-1988*)