Science:Math Exam Resources/Courses/MATH105/April 2012/Question 05 (b)/Solution 1

We write H as a sum of F and G:

$H(x,y)=\int _{y}^{0}\ln(2+\sin t)dt+\int _{0}^{x}\ln(2+\sin t)dt=G(y)+F(x)$.

Then

$\displaystyle H_{x}={\frac {\partial }{\partial x}}(F(x)+G(y))=F'(x)+0=\ln(2+\sin(x))$

and similarly

$\displaystyle H_{y}={\frac {\partial }{\partial y}}(F(x)+G(y))=0+G'(y)=-\ln(2+\sin(y))$

where we used the knowledge about the derivatives F'(x) and G'(y) from part (a). Finally, all that is left to do is to plug in the given values to find that

$\nabla H\left({\frac {\pi }{2}},{\frac {\pi }{2}}\right)=\left\langle F'\left({\frac {\pi }{2}}\right),G'\left({\frac {\pi }{2}}\right)\right\rangle =\langle \ln 3,-\ln 3\rangle$