Let so that Find the linear approximation of on a neighborhood of Don't worry about the invertibility, that was another question which I solved, but now I need help with this one, I don't get how to solve it.
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By the chain rule, we get that .
Okay... but, I don't know how to relate it to find the linear approximation. If you could help me a bit more.
It's pretty straightforward: Since that relation gives that the derivatives involved are invertible we get
Oh, now I get, so I just need to compute the inverse but do I need to multiply it by a vector or something?
The linear approximation to g(x, y) at (x_0, y_0) is g(x_0, y_0)+ D_g(x_0, y_0)(x, y) where the last term is the product of the matrix D_g(x_0, y_0) with the vector (x, y).
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