(1)• Theorem 10.5.1 The Chain Rule (I) If z = f (x(t), y(t

• Theorem 10.5.1 The Chain Rule (I)

If z = f (x(t), y(t)), where x(t) and y(t) are differentiable and f (x, y) is a differentiable function of x and y, then

dz dt = d

dt[f (x(t), y(t))] = ∂f

∂x(x(t), y(t))dx dt +∂f

∂y(x(t), y(t))dy dt.

• Theorem 10.5.2 Chain Rule (II)

Suppose that z = f (x, y), where f is a differentiable function of x and y and where x = x(s, t) and y = y(s, t) both have first-order partial derivatives. Then we have the chain rules:

∂z

∂s = ∂z

∂x

∂x

∂s +∂z

∂y

∂y

∂s

∂z

∂t = ∂z

∂x

∂x

∂t +∂z

∂y

∂y

∂t.

• Theorem 10.7.2 Second Derivatives Test

Suppose that f (x, y) has continuous second-order partial derivatives in some open disk containing the point (a, b) and that fx(a, b) = fy(a, b) = 0. Define the discriminant D for the point (a, b) by

D(a, b) = fxx(a, b) fyy(a, b)− [ fxy(a, b)]².

(1) If D(a, b) > 0 and fxx(a, b) > 0, then f has a local minimum at (a, b).

(2) If D(a, b) > 0 and f_xx(a, b) < 0, then f has a local maximum at (a, b).

(3) If D(a, b) < 0, then f has a saddle point at (a, b).

(4) If D(a, b) = 0, then no conclusion can be drawn.

• Theorem 10.7.3 Extreme Value Theorem (I)

Suppose that f (x, y) is continuous on the closed and bounded region R ⊂ R². Then f has both an absolute maximum and an absolute minimum on R.

Further, the absolute extrema must occur at either a critical point in R or on the boundary of R.

• Theorem 10.8.1 Method of Lagrange Multiplier

Suppose that f (x, y) and g(x, y) are functions with continuous first partial derivatives and ∇g(x, y) 6= 0 on the surface g(x, y) = 0. Suppose that either

(1) the minimum value of f (x, y) subject to the constraint g(x, y) = 0 occurs at (x₀, y₀); or

(2) the maximum value of f (x, y) subject to the constraint g(x, y) = 0 occurs at (x0, y0).

Then ∇f(x0, y₀, z₀) = λ∇g(x0, y₀, z₀) for some constant λ.

• Definition 11.8.1 Jacobian of a transformation

∂(x, y)

∂(u, v) =

¯¯¯¯

¯

∂x

∂u

∂x

∂v

∂y

∂u

∂y

∂u

¯¯¯¯

¯

• Theorem 11.8.1

Suppose that the region S in the uv-plane is mapped onto the region R in the xy-plane by the one-to-one transformation T defined by x = g(u, v) and y = h(u, v), where g and h have continuous first partial derivatives on S. If f is continuous on R and the Jacobian ∂(x, y)

∂(u, v) is nonzero on S, then

¨

R

f (x, y)dA =

¨

S

f (g(u, v), h(u, v))¯¯

¯¯∂(x, y)

∂(u, v)

¯¯¯¯dudv.