Partial Derivatives

Def: A function z = f(x,y) of 2 variables x,y is a rule that assigns to each pair (x,y) a single value for z. x,y are independent variables while z is a dependent variable. The domain is a subset of .

Def: The level curves of a function f of two variables are the curves with equations f(x,y) = k, where k is a constant (in the range of f).

Def: The limit of a function of 2 variables is defined as: . We can get arbitrarily close to the point (a,b). If we get close to (a,b) no matter how we approach it the function function should approach L.

Technique 1: If the limit appears to exist after approaching the point from different methods. We can convert x & y to polar cords and try to determine the limit this way. So let & let .

Def: A function of two variables is called continuous at (a,b) if We say f is continuous on D if f is continuous at every point (a,b) in D.

Def: Partial derivative with respect to x is represented as:

Def: Partial derivative with respect to y is represented as:

The geometric representation of a partial derivative with respect to x is just the slope of the solid where y is constant if it is with respect to y then x is taken as constant.

Theorem: Suppose are defined on a disk D centered at (a,b) and are continuous at every point in D. Then .

Harmonic Functions:

Wave Equation:

Tangent Plane:

Linear Approximations (Linerazation): . Note: This is the start of a Taylor Series.

Theorem: A function f(x,y) is differentiable at if both partial derivatives exist at this point and they are continuous on a disk, centered at .

Differential: Note: dz is the linear part of the increment of the function f(x,y).

Chain Rule: for the chain rule follows that:

&

Directional Derivatives: Measure the rate of change along any direction. A Directional derivative of f(x,y) at in the direction of a unit vector u with coordinates is

provided this limit exists.

Theorem: then

Gradient Vector:

Theorem: If is a differentiable function in x & y. Then the maximum value of is it occurs in the direction of .

Equation of the Tangent Plane:

Normal Line: Parametric Equations:

Def: A function z=f(x,y) has a local maximum at (a,b) if f(x,y) f(a,b) for all (x,y) that are near by ((a,b),f(a,b)).

Def: A function z=f(x,y) has a local minimum at (a,b) if f(x,y) f(a,b) for all (x,y) that are near by ((a,b),f(a,b)).

IF at (a,b) then point (a,b) is a critical point.

Theorem: The second derivative test: suppose (a,b) is a critical point then.

If D(a,b) > 0 & then we have a local min at (a,b)

If D(a,b) > 0 & then we have a local max at (a,b)

If D(a,b) < 0 & then we have neither a local min at (a,b) or a local max at (a,b) instead we have a saddle point.

If D(a,b) = 0 then we cannot determine anything.

D(a,b) is the discriminant:

Lagrange Multipliers: Are another way to determine the min and max of a function. For a function f(x,y) we can subject it to the constraints of g(x,y) = 0 then the min and max of f(x,y) will be the points where f(x,y) is tangent to one of the level curves of g(x,y) ie.

Example: Use Lagrange multipliers to find the coordinates of the point (x, y, z) on the plane z = 3 x + 2 y + 2 which is closest to the origin.

Let f(x,y,z) = and be under the constraint g(x,y,z)= 3 x + 2 y –z + 2

Then then so we have 3 equations:

1. 2x=

2. 2y= 3. 2z=

4. Therefore:

Insert into equation g(x,y,z): 3(-3z)+2(-2z)-z+2=0 solve for z then z = use to solve for x and y

and you get x = & y=