Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Calculus, probability, linear algebra, discrete mathematics, logic and proof, and models of computation / 2 / a / i

Assume all partial derivatives needed below are continuous. The multivariable chain rule is

f_{x} = g_{u} u_{x} + g_{v} v_{x},

g

are evaluated at

(u (x, y), v (x, y))

f_{xx} = (g_{uu} u_{x} + g_{uv} v_{x}) u_{x} + g_{u} u_{xx} + (g_{vu} u_{x} + g_{vv} v_{x}) v_{x} + g_{v} v_{xx} .

g_{vu} = g_{uv}

, so

f_{xx} = g_{u} u_{xx} + g_{v} v_{xx} + g_{uu} u_{x}^{2} + 2 g_{uv} u_{x} v_{x} + g_{vv} v_{x}^{2} .

Differentiating

f_{x}

with respect to

y

instead gives

f_{x y} = g_{u} u_{x y} + g_{v} v_{x y} + g_{uu} u_{x} u_{y} + g_{uv} (u_{x} v_{y} + v_{x} u_{y}) + g_{vv} v_{x} v_{y} .

 Ancestors (12)

There are no discussions about this article yet.

There are currently no matching articles.