Real Valued Functions on R^3 - Chain Rule ....?

[Math Amateur]

I am reading Barrett O'Neil's book: Elementary Differential Geometry ...

I need help to get started on Exercise 4(a) of Section 1.1 Euclidean Space ...

Exercise 4 of Section 1.1 reads as follows:View attachment 5186Can anyone help me to get started on Exercise 4(a) ...

I would guess that we need the chain rule for multiple variables but how do we formulate the dependencies of the functions involved and what is the correct form of the chain rule to use ...

Help will be appreciated ...

Peter

 

[Deveno]

I would use the following function:

$g: \Bbb R^3 \to \Bbb R^3$

given by $g(x,y,z) = (g_1(x,y,z),g_2(x,y,z),g_3(x,y,z))$

Then $f= h \circ g$ and the chain rule gives:

$Df(\mathbf{p}) = Dh(g(\mathbf{p}))Dg(\mathbf{p})$

The LHS will be a 1X3 matrix, and the RHS will be the matrix product of a 1X3 matrix and a 3X3 matrix.

Each ENTRY of the LHS (which will be your partials $\dfrac{\partial f}{\partial x_i}$) will be a dot product.