## Thomas' Calculus 13th Edition

To evaluate $F_{yx}$ we use Fubini's Theorem to rewrite $F(x, y)$ as $\int^y_c\int^x_af(u,v)du dv$ and make a similar argument. The result is again $f(x,y)$.
Since $f$ is continuous on $R$, for fixed $u$ $f(u, v)$ is a continuous function of $v$ and has an antiderivative with respect to $v$ on $R$, call it $g(u,v)$ . Then $\int^y_cf(u,v)dv$ = $g(u,v)-g(u,c)$ and $F(x,y)$=$\int^x_a\int^y_cf(u,v) dv du$ = $\int^x_a(g(u,y)-g(u,c))du$ $F_x$=$\frac{\partial}{\partial x}\int^x_a(g(u,v)-g(u,c)) du$ = $g(x,y)-g(x,c)$ Now taking the derivative with respect to $y$,we get $F_{xy}$=$\frac{\partial}{\partial x}(g(x,y)-g(x,c))$ To evaluate $F_{yx}$ we use Fubini's Theorem to rewrite $F(x, y)$ as $\int^y_c\int^x_af(u,v)du dv$ and make a similar argument. The result is again $f(x,y)$.