Unit 15: Divergence and Curl

The Concept

Divergence of vector field [latex]\vec{F}[/latex] is defined as an operation on a vector field that tells us how the field behaves toward or away from a point. Locally, the divergence of a vector field [latex]\vec{F}[/latex] at a particular point [latex]P[/latex] in 2D or 3D is a scalar measure of the “outflowing-ness” of the vector field [latex]\vec{F}[/latex] at point [latex]P[/latex].

Mathematically, we can define divergence as:

• If [latex]\vec{F}=\langle P(x,y), Q(x,y) \rangle[/latex] is a vector field in 2D, and [latex]P_x[/latex] and [latex]Q_y[/latex] both exist, then the divergence of [latex]\vec{F}[/latex] is defined by [latex]div(F) =P_x+Q_y =  \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}[/latex].

• If [latex]\vec{F}=\langle P(x,y,z), Q(x,y,z), R(x,y,z)\rangle[/latex] is a vector field in 3D and [latex]P_x[/latex], [latex]Q_y[/latex], and  [latex]R_z[/latex] all exist, then the divergence of [latex]\vec{F}[/latex] is defined by [latex]div(F) =P_x+Q_y+R_z  =  \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}[/latex].

In other words, [latex]div(F)[/latex] is equal to the rate of change of [latex]P, Q[/latex] or [latex]R[/latex] in each respective direction added together. If we think of this logically and add the rate of change in each direction at a specific point, then we will get the rate of change and direction at that point.

Curl of vector field [latex]\vec{F}[/latex] is denoted as [latex]curl(\vec{F})[/latex], which measures the extent of rotation of the field about a point. Suppose that [latex]\vec{F}[/latex] represents the velocity field of a fluid. Then, the curl of [latex]\vec{F}[/latex] at point [latex]P[/latex] is a vector that measures the tendency of particles near [latex]P[/latex] to rotate about the axis that points in the direction of this vector.

• If [latex]\vec{F}=\langle P, Q \rangle[/latex] is a vector field in 2D, and [latex]P_x[/latex] and [latex]Q_y[/latex] both exist, then the curl of [latex]\vec{F}[/latex] is defined by [latex]Curl(\vec{F}) = (Q_x - P_y)\vec{k} = \langle 0, 0, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \rangle[/latex]

• If [latex]\vec{F}=\langle P, Q ,R\rangle[/latex] is a vector field in 3D, and [latex]P_x, Q_y, and R_z[/latex] all exist, then the curl of [latex]\vec{F}[/latex] is defined by [latex]Curl(\vec{F}) = (R_y-Q_z)\vec{i} + (P_z-R_x)\vec{j} + (Q_x - P_y)\vec{k} = \langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \rangle[/latex]

Divergence and curl are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dimensional versions of the fundamental theorem of calculus. In addition, curl and divergence appear in mathematical descriptions of fluid mechanics, electromagnetism and elasticity theory, which are important concepts in physics and engineering. We can also apply curl and divergence to other concepts we already explored. For example, under certain conditions, a vector field is conservative if and only if its curl is zero.

The Plot

Now, you should engage with the plot below to understand divergence and curl^[1]. Follow the steps below to apply changes to the plot and observe the effects:

1. Fill in a field function.
2. Choose a path.
3. The graph depicted shows the divergence and curl.