Math2111: Chapter 2: Vector fields and the operator ∇. Section 2: Divergence and Curl

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We discuss now derivatives of vector fields. To that end we introduce the del operator ∇. Throughout we assume that the vector field \boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}^3 and the scalar field f:\mathbb{R}^3\to\mathbb{R} are \mathcal{C}^1, i.e. have at least one continuous partial derivative in each coordinate (similarly we write \mathcal{C}^2 if the functions are assumed to have continuous partial derivative of order at least two).

In the following we write vectors either as (a,b,c)\in\mathbb{R}^3 or as a \boldsymbol{\mathrm{i}}+b\boldsymbol{\mathrm{j}}+c\boldsymbol{\mathrm{k}}\in\mathbb{R}^3 (i.e. both have the same meaning).

The del operator ∇

We formally introduce the symbol {}\nabla as

\displaystyle \nabla = \boldsymbol{\mathrm{i}} \frac{\partial}{\partial x} + \boldsymbol{\mathrm{j}} \frac{\partial }{\partial y} + \boldsymbol{\mathrm{k}} \frac{\partial }{\partial z},

which is an operator. That is, it makes sense when it acts or operates on real-valued functions. For instance, if f:\mathbb{R}^3\to \mathbb{R}, then

\displaystyle \nabla f = \boldsymbol{\mathrm{i}} \frac{\partial f}{\partial x} + \boldsymbol{\mathrm{j}} \frac{\partial f}{\partial y} + \boldsymbol{\mathrm{k}} \frac{\partial f}{\partial z},

the gradient of {}f.

Divergence

Definition (Divergence)
The divergence \mathrm{div}\boldsymbol{F}:\mathbb{R}^3\to \mathbb{R} of a vector field \boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}, \boldsymbol{F}=(F_1,F_2,F_3) = F_1 \boldsymbol{\mathrm{i}} + F_2 \boldsymbol{\mathrm{j}} + F_3 \boldsymbol{\mathrm{k}}, is defined by

\displaystyle \mathrm{div} \boldsymbol{F} = \nabla \cdot \boldsymbol{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}.

Vector fields for which \mathrm{div} \boldsymbol{F}=0 are called incompressible.

Example
Let \boldsymbol{F}=(x^2y,xyz,yz^2)=x^2y\boldsymbol{\mathrm{i}}+xyz\boldsymbol{\mathrm{j}}+yz^2\boldsymbol{\mathrm{k}}. Hence we have F_1=x^2y, F_2=xyz and F_3=yz^2. Then the divergence is given by

\displaystyle \begin{array}{rcl} \mathrm{div}\boldsymbol{F} & = & \nabla \cdot \boldsymbol{F} = \frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z} \\ && \\ & = & \frac{\partial}{\partial x} x^2y + \frac{\partial }{\partial y} xyz + \frac{\partial }{\partial z} yz^2 = 2xy+xz+2yz. \end{array}

Curl

Definition (Curl)
The curl \mathrm{curl}\boldsymbol{F}:\mathbb{R}^3\to \mathbb{R}^3 of a vector field \boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}, \boldsymbol{F}=(F_1,F_2,F_3) = F_1 \boldsymbol{\mathrm{i}} + F_2 \boldsymbol{\mathrm{j}} + F_3 \boldsymbol{\mathrm{k}}, is defined by

\displaystyle \begin{array}{rcl} \mathrm{curl} \boldsymbol{F} & = &\nabla \times \boldsymbol{F} \\ && \\ & = & \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}\right)\boldsymbol{\mathrm{i}} + \left(\frac{\partial F_1}{\partial z}- \frac{\partial F_3}{\partial x}\right)\boldsymbol{\mathrm{j}}+\left(\frac{\partial F_2}{\partial x}- \frac{\partial F_1}{\partial y}\right)\boldsymbol{\mathrm{k}}. \end{array}

Vector fields for which \mathrm{curl} \boldsymbol{F}= \boldsymbol{0} are called irrotational.

Note
The formula for the \mathrm{curl} is easiest to remember in operator notation as the determinant of the following matrix:

\displaystyle \mathrm{curl}\boldsymbol{F}= \left|\begin{array}{ccc} \boldsymbol{\mathrm{i}} & \boldsymbol{\mathrm{j}} & \boldsymbol{\mathrm{k}} \\ && \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ && \\ F_1 & F_2 & F_3 \end{array} \right|.

Example
Let \boldsymbol{F}=(x^2y,xyz,yz^2)=x^2y\boldsymbol{\mathrm{i}}+xyz\boldsymbol{\mathrm{j}}+yz^2\boldsymbol{\mathrm{k}}. Hence we have F_1=x^2y, F_2=xyz and F_3=yz^2. Then the curl is given by

\displaystyle \begin{array}{rcl} \mathrm{curl}\boldsymbol{F} & = & \nabla \times \boldsymbol{F} \\ && \\ & = & \left|\begin{array}{ccc} \boldsymbol{\mathrm{i}} & \boldsymbol{\mathrm{j}} & \boldsymbol{\mathrm{k}} \\ && \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial }{\partial z} \\ && \\ F_1 & F_2 & F_3 \end{array} \right| = \left|\begin{array}{ccc} \boldsymbol{\mathrm{i}} & \boldsymbol{\mathrm{j}} & \boldsymbol{\mathrm{k}} \\ && \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial }{\partial z} \\ && \\ x^2y & xyz & yz^2 \end{array} \right| \\ && \\ & = & (z^2-xy, 0, yz-x^2).\end{array}

Laplace operator

Definition (Laplace operator)
Let f:\mathbb{R}^3\to\mathbb{R} with continuous partial derivatives of second order in each variable. Then the Laplace operator acts on {}f as follows

\displaystyle \triangle f= \nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.

Some common identities

Theorem
For any \mathcal{C}^2 vector field \boldsymbol{F} we have

\displaystyle {\rm div}\,{\rm curl}\,\boldsymbol{F} = \nabla \cdot (\nabla \times \boldsymbol{F}) = 0,

that is, the divergence of the any curl is zero.

Hint Expand and use Clairaut’s theorem.

Theorem
For any \mathcal{C}^2 function \boldsymbol{f}:\mathbb{R}^3\to\mathbb{R} we have

\displaystyle {\rm curl}\,{\rm grad}\,\boldsymbol{f} = \nabla \times (\nabla \boldsymbol{f}) = \boldsymbol{0},

that is, the curl of any gradient is the zero vector.

Hint Expand and use Clairaut’s theorem.

Some common identities
Let \boldsymbol{F},\boldsymbol{G},\boldsymbol{H}:\mathbb{R}^3\to\mathbb{R}^3 be vector fields and f,g:\mathbb{R}^3\to\mathbb{R} be scalar fields.

  1. \nabla(f+g) = \nabla f + \nabla g
  2. \nabla(cf) = c\nabla f\, for a constant {}c
  3. \nabla (fg) =f \nabla g + g \nabla f
  4. \nabla (f/g) = \frac{g\nabla f - f \nabla g}{g^2}\, at points where g(\boldsymbol{x}) \neq 0
  5. \mathrm{div}(\boldsymbol{F}+\boldsymbol{G}) =\mathrm{div}\boldsymbol{F} + \mathrm{div} \boldsymbol{G}
  6. \mathrm{curl} (\boldsymbol{F}+\boldsymbol{G}) =\mathrm{curl}\boldsymbol{F}+ \mathrm{curl}\boldsymbol{G}
  7. \nabla(\boldsymbol{F}\cdot\boldsymbol{G}) = (\boldsymbol{F}\cdot\nabla)\boldsymbol{G}+(\boldsymbol{G}\cdot\nabla)\boldsymbol{F}+F \times \mathrm{curl}\boldsymbol{G}+ \boldsymbol{G}\times \mathrm{curl}\boldsymbol{F}
  8. \mathrm{div} (f\boldsymbol{F}) = f \mathrm{div} \boldsymbol{F} + \boldsymbol{F}\cdot \nabla f
  9. \mathrm{div} (\boldsymbol{F} \times \boldsymbol{G}) = \boldsymbol{G} \cdot \mathrm{curl} \boldsymbol{F} - \boldsymbol{F} \cdot \mathrm{curl}\boldsymbol{G}
  10. \mathrm{curl} (f\boldsymbol{F}) =f \mathrm{curl} \boldsymbol{F} + \nabla f \times \boldsymbol{F}
  11. \mathrm{curl} (\boldsymbol{F} \times \boldsymbol{G}) =\boldsymbol{F} \mathrm{div} \boldsymbol{G} -\boldsymbol{G}\mathrm{div}\boldsymbol{F}+(\boldsymbol{G} \cdot \nabla)\boldsymbol{F} - (\boldsymbol{F}\cdot \nabla)\boldsymbol{G}
  12. \mathrm{curl}\mathrm{curl}\boldsymbol{F}=\mathrm{grad}\mathrm{div}\boldsymbol{F}-\nabla^2\boldsymbol{F}
  13. \nabla(\boldsymbol{F}\cdot\boldsymbol{F}) = 2(\boldsymbol{F}\cdot\nabla)\boldsymbol{F}+2\boldsymbol{F}\times (\mathrm{curl}\boldsymbol{F})
  14. \nabla^2 (fg) = f \nabla^2 g + g\nabla^2 f + 2(\nabla f \cdot \nabla g)
  15. \mathrm{div}(\nabla f \times \nabla g) =0
  16. \nabla\cdot (f\nabla g -g\nabla f)=f \nabla^2 g-g\nabla^2 f
  17. \boldsymbol{H}\cdot(\boldsymbol{F}\times\boldsymbol{G})=\boldsymbol{G}\cdot(\boldsymbol{H}\times \boldsymbol{F})=\boldsymbol{F}\cdot(\boldsymbol{G}\times\boldsymbol{H})
  18. \boldsymbol{H}\cdot((\boldsymbol{F}\times \nabla)\times\boldsymbol{G})=((\boldsymbol{H}\cdot\nabla)\boldsymbol{G})\cdot\boldsymbol{F}-(\boldsymbol{H}\cdot\boldsymbol{F})(\nabla\cdot\boldsymbol{G})
  19. \boldsymbol{F}\times(\boldsymbol{G}\times\boldsymbol{H})=(\boldsymbol{F}\cdot\boldsymbol{H})\boldsymbol{G}-\boldsymbol{H}(\boldsymbol{F}\cdot\boldsymbol{G})

Note
Let \boldsymbol{F}=(F_1,F_2,F_3):\mathbb{R}^3\to\mathbb{R}^3 and \boldsymbol{G}=(G_1,G_2,G_3):\mathbb{R}^3\to\mathbb{R}^3 be vector fields, where F_1,F_2,F_3,G_1,G_2,G_3:\mathbb{R}^3\to\mathbb{R} are scalar fields.

  • The expression \boldsymbol{F}\cdot\nabla denotes an operator, in particular \boldsymbol{F}\cdot \nabla\neq \nabla\cdot\boldsymbol{F}. For instance

    \displaystyle (\boldsymbol{F}\cdot \nabla)\boldsymbol{G}=\boldsymbol{F}\cdot(\nabla G_1) \boldsymbol{\mathrm{i}}+\boldsymbol{F}\cdot(\nabla G_2) \boldsymbol{\mathrm{j}} + \boldsymbol{F}\cdot(\nabla G_3)\boldsymbol{\mathrm{k}}.

  • By \nabla^2 \boldsymbol{F} we mean

    \displaystyle \nabla^2\boldsymbol{F}=\nabla^2 F_1\boldsymbol{\mathrm{i}} + \nabla^2 F_2 \boldsymbol{\mathrm{j}}+\nabla^2 F_3\boldsymbol{\mathrm{k}}.

  • Further

    \displaystyle (\boldsymbol{F}\times \nabla)\times \boldsymbol{G} = \boldsymbol{F}\times (\nabla G_1) \boldsymbol{\mathrm{i}}+ \boldsymbol{F}\times (\nabla G_2) \boldsymbol{\mathrm{j}}+\boldsymbol{F}\times (\nabla G_3)\boldsymbol{\mathrm{k}}.

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