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MATH
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Theorems from First-order methods

All theorems from First-order methods.

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The exactness test and potential

Suppose MM and NN have continuous first partial derivatives on an open simply connected domain DD. If

My=∂M∂y=Nx=∂N∂xM_y=\frac{\partial M}{\partial y} =N_x=\frac{\partial N}{\partial x}

throughout DD, then the equation M dx+N dy=0M\,dx+N\,dy=0 has a potential FF on DD and its solution curves satisfy

F(x,y)=C.F(x,y)=C.

Matching partial derivatives guarantees a single-valued potential on a simply connected domain, while holes in the domain can obstruct a global potential.

  1. [1]:Exact differential equation
  2. [2]:Constructing the potential
  3. [3]:Exact equations and level-set geometry

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The exactness test and potential