160 CHAPTER 6. THE DERIVATIVE

Subtracting the first sum on the right from both sides and using the equality of mixedpartials,

∑i

gi,k

(∑

j(cof(Dg))i j, j

)= 0.

Since it is assumed Dg is invertible, this shows ∑ j (cof(Dg))i j, j = 0. If det(Dg) = 0, useLemma 6.11.1 to let gk (x) = g (x)+εkx where εk→ 0 and det(Dg+ εkI)≡ det(Dgk) ̸=0. Then

∑j(cof(Dg))i j, j = lim

k→∞∑

j(cof(Dgk))i j, j = 0 ■

6.12 Exercises1. Here are some scalar valued functions of several variables. Determine which of these

functions are o(v). Here v is a vector in Rn, v = (v1, · · · ,vn).

(a) v1v2

(b) v2 sin(v1)

(c) v21 + v2

(d) v2 sin(v1 + v2)

(e) v1 (v1 + v2 + xv3)

(f) (ev1 −1− v1)

(g) (x ·v) |v|

2. Here is a function of two variables. f (x,y) = x2y+ x2. Find D f (x,y) directly fromthe definition. Recall this should be a linear transformation which results from mul-tiplication by a 1×2 matrix. Find this matrix.

3. Let f (x,y) =(

x2 + yy2

). Compute the derivative directly from the definition. This

should be the linear transformation which results from multiplying by a 2×2 matrix.Find this matrix.

4. You have h(x) = g (f (x)) Here x ∈ Rn, f (x) ∈ Rm and g (y) ∈ Rp. where f,gare appropriately differentiable. Thus Dh(x) results from multiplication by a matrix.Using the chain rule, give a formula for the i jth entry of this matrix. How does thisrelate to multiplication of matrices? In other words, you have two matrices whichcorrespond to Dg (f (x)) and Df (x) Call z = g (y) ,y = f (x) . Then

Dg (y) =(

∂z∂y1

· · · ∂z∂ym

),Df (x) =

(∂y∂x1

· · · ∂y∂xn

)Explain the manner in which the i jth entry of Dh(x) is ∑k

∂ zi∂yk

∂yy∂x j

. This is a review

of the way we multiply matrices. what is the ith row of Dg (y) and the jth column ofDf (x)?

5. Find fx, fy, fz, fxy, fyx, fzy for the following. Verify the mixed partial derivatives areequal.

(a) x2y3z4 + sin(xyz)

(b) sin(xyz)+ x2yz