Let f be a function of two variables that has continuous partial derivatives Suppose u 1 and u 2 are two linearly independent and orthogonal vectors in R 2 , that is, u 1 (unequal) cu 2 for all c in R and u 1 u 2 0 If Du 1 f(x,y) k 1 and Du 2 f(x,y) k 2 , show that Duf(x,y) (k 1 u 1 k 2 u 2 ) u Hint Use the fact that each v in R 2 can be represented as v (u1 v)u1 (u2 v)u2 Explain as best as you can please

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Question: Let f be a function of two variables that has continuous partial derivatives. Suppose u 1 and u 2 are two linearly independent and orthogonal

Let f be a function of two variables that has continuous partial derivatives. Suppose u₁ and u₂ are two linearly independent and orthogonal vectors in R², that is, u₁=/ (unequal) cu₂for all c in R and u₁ u₂=0. If Du₁f(x,y) = k₁ and Du₂f(x,y)=k₂, show that Duf(x,y) = (k₁u₁+ k₂u₂) u

Hint: Use the fact that each v in R²can be represented as v = (u1 v)u1 + (u2 v)u2

Explain as best as you can please.

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