I’m really struggling with this question:( if someone can help with it or even give some tips to start with please, thank you.

Guest Apr 29, 2019

#1**+1 **

\(T(x,y) = xy^2 + x^2 y\\ \text{in general given a function }F(x,y)\\ \nabla F = \left(\dfrac{\partial F}{\partial x},~\dfrac{\partial F}{\partial y}\right)\)

\(a)~\nabla T = \left(y^2+2xy,~2xy+x^2\right)\\ b)~u=Q-P = (2,-2)\\ \dfrac{u}{|u|} = \dfrac{1}{\sqrt{2}}(1,-1)\\ \nabla_u T = \nabla T \cdot \dfrac{u}{|u|} = \left . \dfrac{y^2-x^2}{\sqrt{2}} \right |_{(-1,3)}=4\sqrt{2} \\ c)~\text{The most rapid decrease occurs in the direction of the gradient}\\ \left . \nabla T \right|_{(-1,3)} = \dfrac{(-3,5)}{|(-3,5)|} = \dfrac{1}{\sqrt{34}}(-3,5)\)

.Rom Apr 29, 2019