Irrotational Fields An irrotational field is a vector field in a 2-dimensional plane or a 3-dimensional space that is characterized by the following propert...
Irrotational Fields
An irrotational field is a vector field in a 2-dimensional plane or a 3-dimensional space that is characterized by the following property: the vector field is the gradient of a scalar function.
Scalar Function: A scalar function is a function that assigns a single numerical value to each point in the domain.
Gradient: The gradient of a scalar function is a vector that points in the direction of the steepest ascent of the function.
Irrotational Field: A vector field is irrotational if the curl of the vector field is equal to the zero vector.
Curl of a Vector Field: The curl of a vector field is a measure of the "curvature" or "twist" of the field.
Zero Vector: The zero vector is a vector field that is everywhere zero. An irrotational field is one that is everywhere non-zero.
Examples:
The gradient of a function of two variables is an irrotational field. For example, the gradient of the function f(x, y) = x^2 + y^2 is an irrotational field.
A vector field that is perpendicular to the xy-plane is an irrotational field.
The vector field v(x, y) = (y, -x) is an irrotational field