Chapter 8:Functions Of Several Variables

Question 1

Limits

Answer 1

Limit of a function of several variables has similar properties as those of limit of a function of a single variable.
Theorem:
Assume that M=(x,y)
lim f(M),lim g(M) are finite. Then:
lim (f(M)+g(M)) = lim f(M)+lim g(M) [same goes for (-)]
lim (f(M).g(M)) = limf(M) . limg(M) [same goes for (/)]
Squeeze theorem:
If f(M)≤g(M)≤h(M) when M is close to M0 and
lim f(M) = lim h(M) = L ,then lim g(M) =L as M tend to M0

Question 2

Continuity

Answer 2

The function f(M) is said to be continuous at M0 if limf(M) =f(M0) as M tend to M0.
Note:It’s continuity is basically the same as the one with a single variable.

Question 3

Partial Derivatives

Answer 3

The partial derivative with respect to x is
f′x(x0,y0) = ∂f/∂x = df/dx = lim∆x→0
(f(x0+ ∆x,y0)−f(x0,y0))/∆x
The partial derivative with respect to y is
f′y(x0,y0) = ∂f/∂y = df/dy = lim∆y→0
(f(x0,y0+ ∆y)−f(x0,y0))/∆y
Rule: when differentiating with respect to x,y is regarded as aconstant;when differentiating with respect to y,x is regarded as a constant.

Question 4

Higher Derivatives

Answer 4

It’s mechanics is the same as partial derivatives but you have to differentiate twice.
Ex: f’x=∂f/∂x -> f’‘xx=∂(∂f/∂x)/∂x or f’xy=∂(∂f/∂x)/∂y
Theorem:
Suppose that f is defined on D that contains point (a,b) and its neighborhood + f’‘xy and f’‘yx are continuous at (a,b) and its neighborhood then:
f”xy(a,b)=f”yx(a,b)

Question 5

Tangent Planes and Linear Approximations

Answer 5

The tangent plane equation to the surface S at the point P(x0,y0,z0) is:
z-z0=f’x(x-x0)+f’y(y-y0)
Linear Approximations:
f(x,y)~f(x0,y0)+fx(x-x0)+fy(y-y0)
Or for small [∆x] or [∆y]
then f(x0+∆x,y0+∆y)~f(x0,y0)+fx∆x+fy∆y

Question 6

Differentials

Answer 6

Definition:
the function z is called differentiable if z can be expressed in the form below:
dz=df= f’x∆x+f’y∆y
Theorem:
If z=f(x,y) has continuous derivatives fx and fy then f is differentiable.
Definition of differential on a higher order:
(d^2)z=(d^2)f=fxx(x,y)dx^2+2fxy(x,y)dxdy+fyy(x,y)dy^2
Taylor theorem applications for higher total differential:
Let f(x,y) have continuous partial derivatives up to order n+1, then for n=2 we have the explicit expansion as follows:
f(x,y)= f + f’x(x-x0) + f’y(y-y0) + (1/2).(f’‘x.(x-x0)^2 + 2f’‘xy(x-x0)(y-y0) + f’‘yy(y-y0)^2) + R2(x-x0)(y-y0)
The chain rule 1:
Suppose that z=f(x,y) is differentiable, where x=x(t) and y=y(t) are both differentiable functions of t. Then z is a differential function of t and
z=z’x.x’(t)+z’y.y’(t)
The chain rule 2:
Suppose that z=f(x,y) is a differentiable function of x,y where x=x(u,v) and y=y(x,y) are both differentiable functions of u and v. Then z is a differential function of u and v and
z’x=z’u.u’x+z’v.v’x
z’y=z’u.u’y+z’v.v’y

Question 7

Directional Derivatives

Answer 7

Definition: The directional derivative z=f(x,y) at (xo,yo) in the direction given by the unit vector u=(a,b):
Duf(xo,yo) = lim {f(xo+yo+ hu) - f(xo,yo)}/h
=lim (f(xo+ha,yo+hb)-f(xo,yo))/h as h tend to 0.
MEANING: Duf shows the rate of change of z at (xo,yo) in the direction of the unit vector u(a,b)
REMEMBER: f(x,y) has to be a differentiable function to be able to use the equation above.

Question 8

Gradient Vector

Answer 8

∆f(x,y)=gradf=(f’x,f’y)=f’x.i+f’y.j where i,j are the unit vectors of the coordinate axes.
Extended Directional Derivatives: Duf=gradf.u
Theorem: The directional derivative have the maximum value when u has the same direction as the gradient vector f.

Question 9

Extreme values

Answer 9

Necessary condition
(First derivative test): If z(x,y) has a local max or min at Mo and there exists z’x,z’y,then z’x=z’y=0
Mo is called a stationary point.
(Second derivative test): Suppose the second order partial derivatives of z are continuous in B(Mo;e) and z’x=z’y=0. => A=z’‘xx(Mo);B=z’‘xy(Mo);C=z’‘yy(Mo);∆=B^2-AC
IF ∆<0 then A>0 will attain a local min at Mo
A<0 will attain a local max at Mo
∆>0 then z does not attain a local extreme value at Mo which means Mo is a saddle point.