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

 Apr 29, 2019
 #1
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\(T(x,y) = xy^2 + x^2 y\\ \text{in general given a function }F(x,y)\\ \nabla F = \left(\dfrac{\partial F}{\partial x},~\dfrac{\partial F}{\partial y}\right)\)

 

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

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 Apr 29, 2019
 #2
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This is the second time you answer my questions, and I’m more then grateful for your help. Thank you so much, it makes more sense now! 

Guest Apr 29, 2019

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