Vector fields can also be defined on differentiable manifolds. Differentiable manifolds are similar spaces to Euclidean space but they are measured on small scales and may be a complex structure on big scales. A vector field can be defined as a function for both two and three dimensional space. Let’s consider that a two dimensional space has two points (a, b) on it then vector field on this space will be a function R →> that is assigned to each point (a, b) which is a two dimensional vector R→> (a, b).
Vector field is represented in notation as shown below:
R→> (a, b) = M (a, b) i^ + N (a, b) j^,
Where M, N are called Scalar functions.
Let’s consider following examples to understand how to draw vector fields.
R→> (a, b) = - b i^ + a j^,
Let a = 1 / 2 and b = 1 / 2,
R→> (1 / 2, 1 / 2) = - 1 / 2i^+ 1 / 2j^,
When a = 3 / 2 and b = 1 / 4 then,
R→> (3/ 2, - 1 / 4) = - 1 / 4i^ + 3/ 2j^,
In above cases, for point (1 / 2, 1 / 2), we will plot the vector field - 1 / 2i^ + 1 / 2j^ and similarly for all other points we will plot the vector fields.