The Implicit Function Theorem---Part 1

Equations in two variables

Solving Systems of EquationsEventually, we will ask: “Under what circumstances can we solve a system of equations in which there are more variables than unknowns for some of the variables in terms of the others?”

Other times…..(Non-linear Systems!)

1 2 3 1 2

1 3 1 2

1 2 3 1 2

2 2 2 11 04 2 5 0

5 0

y y y x xy y x xy y y x x

1 2 3 1 2

1 3 1 2

1 2 3 1 2

2 2 2 11 04 2 5 0

5 0

y y y x xy y x xy y y x x

11 2

3 4 42 2

31 2 1 2 1

1 2 3 1 2

2 4 3sin( ) 0

cos( ) cos( ) 05 0

y y x x y

y x x y xy y y x x

Sometimes we can….(Linear Systems!)

“Solving” Systems of Equations

The existence of a “nice” solution

vs

Actually finding a solution

For now…..Equations in two variables: Can we solve for one variable in terms of the other?

x

3 2

22

2 3 e 2

3 sin( ) 1

2 ln 01

y x x

x y yxy y

x

cos( ) 3 sin( ) 4 0x y x y

Linear equations: Easy to solve.

Some non-linear equations Can be solved analytically.

Can’t solve for Either variable!

2 4 0y x

Some observationsx

x

2 3 e 2or

2 3 e 2 0

y x x

y x x

x( , ) 2 3 e 2F x y y x x

x( , ) 2 3 e 2 0F x y y x x

x1 2 3 e2

y x x

x1( ) 2 3 e2

y g x x x

2F ::g

R RR R

Further observations( , ) cos( ) 3 sin( ) 4 0F x y x y x y cos( ) 3 sin( ) 4 0x y x y

The pairs (x,y) that satisfy the equation F(x,y)=0 lie on the 0-level curves of the function F.

That is, they lie on the intersection of the graph of F and the horizontal plane z = 0.

Taking a “piece”The 0-level curves of F.

Though the points on the 0-level curves of F do not form a function, portions of them do.

Summing up

Solving an equation in 2 variables for one of the variables is equivalent to finding the “zeros” of a function from

Such an equation will “typically” have infinitely many solutions. In “nice” cases, the solution will be a function from

2F : R R

:g R R

More observations

The previous diagrams show that, in general, the 0-level curves are not the graph of a function.

But, even so, portions of them may be. Indeed, if the function F is “well-behaved,” we

can hope to find a solution function in the neighborhood of a single known solution.

Well-behaved in this case means differentiable (locally planar).

),( yxfz

0),( yxf

x

y

Consider the contour line f (x,y) = 0 in the xy-plane.Idea: At least in small regions, this curve might be described

by a function y = g(x) . Our goal: To determine when this can be done.

x

y(a,b)

(a,b)

y = g(x)

Start with a point (a,b) on the contour line, where the contour is not vertical.

In a small box around (a,b), we can hope to find g(x).

(What if the contour line at the point is vertical?)

If the contour is vertical. . . We know that y is not

a function of x in any neighborhood of the point.

What can we say about the partial of F(x,y) with respect to y?

Is x a function of y?x

y (a,b)

Other difficult places: “Crossings”

If the 0-level curve looks like an x. . .

We know that y is not a function of x and neither is x a function of y in any neighborhood of the point.

What can we say about the partials of F at the crossing point?

(Remember that F is locally planar at the crossing!)