Calculus Small Changes, Big Ideas

Section 17.5 Potential functions

• If \(\phi\) is a scalar function and \(\nabla\phi=\bv F\text{,}\) we say that \(\phi\) is a potential function of \(\bv F\text{,}\) and that \(\bv F\) is a gradient field.

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  In this case, the vector field is completely determined by a single function.

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• The gradient \(\nabla \phi\) points in the direction of greatest increase of \(\phi\text{.}\) As a result, the vectors of \(\bv F\) are always perpendicular to the level curves (or surfaces) of \(\phi\text{.}\)

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• The Laplacian operator is defined by

  \begin{equation*} \nabla^{2} \phi = \nabla\cdot\nabla \phi = \pdv[2]{\phi}{x}+\pdv[2]{\phi}{y}+\pdv[2]{\phi}{z}. \end{equation*}

  Functions satisfying \(\nabla^2 \phi = 0\) are called harmonic functions.

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• Harmonic functions arise naturally in systems at equilibrium, such as steady-state heat flow, electrostatics, and incompressible fluid flow.

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• In two dimensions, a harmonic function \(\phi\) often admits a companion function \(\psi\text{,}\) called its harmonic conjugate, such that their level curves form orthogonal families.

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• When \(\phi\) and \(\psi\) are related in this way, we can combine them into a complex function

  \begin{equation*} \Omega(z) = \phi(x,y) + \ii \psi(x,y), \end{equation*}

  called a complex potential.

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  If this function is holomorphic, then the corresponding vector field has a highly constrained structure, satisfying the Cauchy–Riemann equations.

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• If \(\bv F(x,y) = (u(x,y), v(x,y))\) represents the velocity field of an ideal fluid (incompressible and irrotational), then the associated complex function encodes both the potential and the flow of the system.

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• Applications:

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  Application	Level curves of \(\phi\)	Level curves of \(\psi\)
  electrostatics	equipotential curves	lines of force
  gravitation	equipotential curves	lines of force
  fluid flow	equipotential curves	streamlines of flow
  heat flow	isotherms	lines of heat flux

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