Reconstructing from Partials

Integrability Condition

• $f$ exists if all mixed partial derivatives are equal
• that means there are $\frac{n(n-1)}{2}$ conditions (where $n$ is the number of dimensions)

2 Var Steps

1. Integrate $f_x^{\prime }\to f$
   • Use $g(y)$ for $C$
2. Set $\frac{\partial }{\partial y}f=f_y^{\prime }$ (given)
3. Solve for $g^{\prime }(y)$
4. $g(y)=\int g^{\prime }(y)\text{d}y$
5. Plug $g(y)$ into the $f$ from 1

3 Var Steps

1. $f=\int f_x^{\prime }$
   • Use $g(y,z)$ for $C$
2. Find $g(y,z)$
   1. Set $\frac{\partial }{\partial y}f=f_y^{\prime }$ (given)
   2. Solve for $g_y^{\prime }(y,z)$
   3. $g(y,z)=\int g_y^{\prime }(y,z)\text{d}y$
      • Use $h(z)$ for $C$
3. Find $h(z)$
   1. Set $\frac{\partial }{\partial z}f=f_z^{\prime }$ (given)
   2. Solve for $h^{\prime }(z)$
   3. $h(z)=\int h^{\prime }(y)\text{d}y$
4. Plug $h(z)$ into $g(y,z)$ into $f$