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Calculus III Hughs-Hallett

Math 232 A,B

Br. Joel Baumeyer

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Multivariable Calculus A function in Three Dimensional

Space (3-D), z = f(x,y): is a function with two independent

variables; it is still a rule that assigns for each of the two independent variables, x and y, one and only one value to the dependent variable z.

e.g. h(x,t) = 5 + cos(0.5x - t) pg 4

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3-D Graphing Basics x coordinate measures the distance from the

yz-plane whose name is: x = 0 y coordinate measures the distance from the

xz-plane whose name is: y = 0 z coordinate measures the distance from the

xy-plane whose name is: z = 0 distance:

222 )()()( czbyaxd

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Graphs of Functions in 2 Variables

The graph of a function of two variables, f, is the set of all points (x,y,z) such that z = f(x,y).

The domain of such a function is is a subset of points in the real Euclidean plane. In general, the graph of a function of two variables is a surface in 3-space.

In particular, the graph of a linear function in 3-space is a plane: ax + by + cz = d.

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Section of a Graph of a Function in 3-D For a function f(x,y), the function

we get by holding x fixed and letting y vary is called a section of f with x fixed. The graph of the section of f(x,y) with x = c is the curve, or cross-section, we get by intersecting the graph of f with the plane x = c. We define a section of f with y fixed similarly.

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Contour Lines (or Level Curves)

Contour lines, or level curves, are obtained from a surface by slicing it by horizontal planes.

If z = f(x,y) then for some value c, where z = c, c = f(x,y) is a level curve for the function.

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Equation of a Plane in 3-D

If a plane has slope m in the x direction, slope n in the y direction, and passes through the point:

then its equation is:

if we write:

then: f(x,y) = c + mx + ny

),,( 000 zyx

)()(),( 000)00 yynxxmzyxfz 000 nymxzc

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Level Surfaces

A function of three variable f(x,y,z), is represented by a family of surfaces of the form: . f(x,y,z) = c, each of which is called a level surface.

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3-D 4-D

A single surface representing a two-variable function: z = f(x,y) can always be thought of a one member of the family of level surfaces representing a three variable function: G(x,y,z) = f(x,y) - z.

The graph of z = f(x,y) is the level surface of G = 0.

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Limits and Continuity The function f as a limit at the point

(a,b), written:

if the difference |f(x,y) - L| is as small as we wish whenever the distance from the pint (x,y) to the point (a,b) is sufficiently small, but not zero.

A function f is continuous at a point if

A function is continuous if it is continuous at each point of its domain

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),(lim),(),(

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