Gradient Operator (∇) is a mathematical operator used with the scalar function to represent the gradient operation. This article discusses the representation of the Gradient Operator in different coordinate systems i.e. Cartesian, Cylindrical and Spherical.
What is the Gradient?
The gradient of the scalar function is a vector representing both the magnitude and direction of the maximum space rate (derivative w.r.t. spatial coordinates) of increase of that function/field.
In simple words, it is like the counterpart of the differentiation in multivariable functions. As in simple function, the differentiation gives the slope, the gradient in the multivariable function gives the maximum change (magnitude and direction).
The gradient of the scalar function is a vector field or a vector. The direction of the gradient vector points always in the direction of the maximum rate of change of function.
Representation of the Gradient Operator
Let the scalar function or field is ψ. Assume that we are required to apply the gradient operation on this ψ. This would be represented as-
The following points are worth noting while applying the gradient operator: –
- The answer would be a vector quantity.
- There must not be (•) or (×) between the ∇ and ψ.
Let V be the scalar field whose gradient to be calculated,
Gradient Operation in Cartesian Coordinates
Gradient Operation in Cylindrical Coordinates
Gradient Operation in Spherical Coordinates
Suggested Read:- What is the Gradient of a Scalar Field?