Multivariable Calculus: Gradient, Divergence, and Curl

The del operator is a key part of the gradient, the divergence, and the curl. The del operator uses partial derivatives and unit vectors. The symbol for the del operator is an inverted triangle.

The del operator is treated like a vector.

The expression for the del operator in Cartesian coordinates is given in the intro image.

A gradient shows the difference in the value of a field. A simple case is the temperature gradient as one moves from one place to another. Connecting points of like temperature gives a contour image of points of like temperature. Another example is a map with contours showing points of given elevations. The gradient is the distances between such contour lines of equal elevation. The gradient is the distance for a given change of a functions value.

In physics forces often change value depending on location, or how fast the magnitude of the is changing. The gradient is the distance between surfaces of like values. Here the contour lines must be replaced with surfaces since a force field often extends in three dimensions.

The gradient is defined as the rates of change in the field in three dimensions. This is in accordance with the del operator being applied to a function.

The mathematical expression for the gradient of a function, f(x,y,z), in a vector field is by applying the del operator to the function, <∂ f(x,y,z)∂x, ∂ f(x,y,z)∂y, ∂ f(x,y,z)∂z>. This shows the changes in each of the three orthogonal directions of the Cartesian coordinate system.

A vector field has vector components of a vector V of <V_x , V_x, V_z>. Both the divergence and curl are defined using a vector field.

The divergence of a vector field is the scalar value that represents the flux. An outward flux is said to be positive. An inward flux is said to be negative.

As an example, consider the velocity of a fluid in a pipe. If the pipe gradually narrows the fluid must speed up to get through the constriction. The velocity vector field has an outward flux as the fluid nears the constriction. After passing the constriction the pipe gradually widens. The fluid flow slows. In the slowing of the flow the flux of the velocity field is negative.

The divergence is formally defined as the scalar product of the del operator and the vector function.

The expression of the divergence in a vector field is given by <∂/∂x, ∂/∂y, ∂/∂z> ∙ V. This is equal to ∂V_x/∂x + ∂V_x/∂y + ∂V_z/∂z.

The curl gives the magnitude and direction of rotation in an infinitesimal distance from a point in the vector field. The curl is a vector where the direction is the direction of the axis of rotation and the magnitude is the rate of rotation.

The method for calculating the curl is to take the cross product of the del operator with the vector field.

Curl = del × F = <∂V_y/∂y - ∂V_z/∂z, ∂V_z/∂z - ∂V_x/∂x, ∂V_x∂x - ∂V_y∂y>

In physics vector fields are often encountered with cylindrical symmetry, such as the electrical and magnetic fields near a long conducting wire. Spherical vector fields also occur, such as the gravitation field near a sphere such as a celestial body.

Cylindrical coordinates are (r, φ, z). The del operator for cylindrical coordinates is given by the expression <∂/∂r, (1/r)∂/∂φ, ∂/∂z>.

Spherical coordinates are (r, θ, φ). The del operator for spherical coordinates is given by the expression <∂/∂r, (1/r)∂/∂θ, (1/r sinθ)∂/∂φ>.

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