Gradient The gradient of a function is a vector containing the partial derivatives of the function with respect to each variable. In other words, it gives u...
Gradient
The gradient of a function is a vector containing the partial derivatives of the function with respect to each variable. In other words, it gives us the rate of change of the function in each direction.
Examples:
The gradient of a function f(x,y) with respect to x is ∂f/∂x.
The gradient of a function f(x,y,z) with respect to x, y, and z is ∂f/∂x, ∂f/∂y, and ∂f/∂z.
Divergence
The divergence of a vector field is a scalar quantity that indicates the rate of change of the vector field at each point. In other words, it tells us how the vector field is pointing at each point.
Examples:
The divergence of a vector field along the vector field itself is equal to the magnitude of the vector field.
The divergence of a vector field at a point is equal to the rate of change of the vector field at that point.
Curl
The curl of a vector field is a scalar quantity that measures the rate of change of the vector field's circulation around a closed curve. In other words, it tells us how the vector field is twisting around the curve.
Examples:
The curl of a vector field around the circle x^2 + y^2 = 1 is 0.
The curl of a vector field around the curve y = x^2 is 2x