CALCULUS 3: MATH 281
Vector-Valued Functions
January 14th, 2016
Mrs. McWilliam & Dr. Kounta (COB)
Outline
Syllabus
Recall: 3 Dimensional Coordinate System
Vector-Valued Functions
Arc Length in R 3
Motion in Space
i. Position
ii. Velocity and Acceleration
Curvature
Tangent and Normal Vectors
Tangential and Normal Component of Acceleration
Parametric Surfaces
THREE-DIMENSIONAL
CARTESIAN
COORDINATE SYSTEM
Please say you remember
Three Dimensional Coordinate System
Calculus III is all the concepts learnt in Calculus I (and
techniques taught in Calculus II) extended to the study
of functions in two or more variables i.e. Calculus III is
Calculus I in in three dimensions.
Your spatial visualization skills needs to be in tune!!!
The 1-Dimensional Coordinate System
The 1-D coordinate system is denoted by R.
x = 3 in R
A number line can be viewed as a 1-D coordinate system
The 2-Dimensional Coordinate System
The 2-D coordinate system is often denoted by R2
x = 3 in R2
(x,y) (abscissa, ordinate)
Cartesian (Rectangular) Coordinate System xy plane
R x R = R2 = {(x,y)| x,y R}
The n-Dimensional Coordinate System
The 3-D coordinate system is often denoted by R3
2
the 2-D coordinate system is often denoted by R
the 1-D coordinate system is denoted by R.
n dimensional
A general n dimensional coordinate system is
often denoted by Rn .
The 3-Dimensional Coordinate System
Standard Position
xy-plane
yz-plane
xz-plane
xy-plane
3D view
xy-plane & xz-plane
3D view
xy-plane, xz-plane & yz-plane
3D view
The Coordinate Planes Divide Space into
Eight Octants
Visualizing 3D
The wall on your left is in the xz-plane.
The wall on your right is in the yz-plane.
The floor is in the xy-plane
Plotting points in Space
To locate the point (x, y, z), we start at the origin O and
proceed as follows:
First, move x units along the
x-axis.
Then, move y units
parallel to the y-axis.
Finally, move z units
parallel to the z-axis.
Ordered tripple
Locating points in Space
3D view of those same points
A better view
An even better view
Locating a Point in Space
Called a rectangular
coordinate system
because points
in space determine
rectangular boxes
To locate a point in space start with the origin move along
the x-axis, move parallel to the y- axis, and then parallel
to the z-axis. R x R x R = R 3 = {(x,y,z)| x,y,z R}
Drawing
Practice:
(x, 0, z)
z = constant
(0, 0, z)
(0, y, z)
P(x, y, z)
(0, y, 0)
y
y = constant
(x, 0, 0)
(x, y, 0)
x = constant
The 3-Dimension coordinate system
3D: Point located in space
2D: Point located on a plane
1D: Point located on a line
Signs of coordinates in each octant
(x,y,z)
Equations
When an equation is given, we must understand from
the context whether it represents either:
A
curve in R2
surface (sometimes a curve) in R3
Equations in R
Equations in R
The second variable is free to take any value since there were no
restrictions on that variable
Equations in R
Equations in R
Equations in R
Equations in R
Equations in R
Consider:
x = 3 in R3
Equations in R
3-incomplete
Equations in R
Equations in R
Equations in R
3-incomplete
Equations in R
Equations in R
3-incomplete
Equations in R
3-incomplete
Equations in R
Equations in R
Equations in R
Graph y = 2x 3 in R2 and R3.
3-incomplete
Equations in R
Equations in R
Equations in R
Equations in R
Parabolic cylinder
Coordinate System
Basic Coordinate System
We say that Q sits in the xy-plane. The xyplane corresponds to all the points which have
a zero z-coordinate. We can also start
at P and move in the other two directions as
shown to get points in the xz-plane (this
is S with a y-coordinate of zero) and the yzplane (this is R with an x-coordinate of zero).
Collectively, the xy, xz, and yz-planes are
sometimes called the coordinate planes.
The point Q is often referred to as the
projection of P in the xy-plane. Likewise, R is
the projection of P in the yz-plane and S is the
projection of P in the xz-plane.
Projections
Projections
Distance Formula in 3D
Midpoint Formula in 3D
A Quick Example
Find the distance between the points P(2, 3, 1)
and Q(4, 1, 5), and find the midpoint of the line
segment PQ.
d P, Q 2 17
M 1,1,3
Can we verify these answers with a graph?
Natural extensions from R to R
Most Formulas that you are use to working with in
R 2 have natural extensions in R 3
Equation of a Sphere
Definition of a Circle
Circle: the set of all points in a plane that lie a fixed
distance from a fixed point.
Definition of a Circle
Sphere: the set of all points that lie a fixed distance
from a fixed point.
fixed distance = radius fixed point = center
Recall the standard equation of a circle???
x h y k
2
Equation of a Sphere
A point P (x, y, z) is on a sphere with center (h, k, l ) and
radius r if and only if
x h y k z l
2
Quick Example: Write the equation for the sphere with its center
at (8, 2, 1) and radius 4 3 .
x 8 y 2 z 1
2
48
How do we graph this sphere???
Graphing a Sphere
Planes and Other Surfaces
We have already learned that every line in the Cartesian
plane can be written as a first-degree (linear) equation in two
variables; every line can be written as
Ax By C 0
How about every first-degree equation in three variables???
They all represent planes in Cartesian space!!!
Planes and Other Surfaces
Equation for a Plane in Cartesian Space
Every plane can be written as
Ax By Cz D 0
where A, B, and C are not all zero. Conversely, every
first-degree equation in three variables represents a
plane in Cartesian space.
Guided Practice
Sketch the graph of
12 x 15 y 20 z 60
Because this is a first-degree equation, its graph is a plane!
Three points determine a plane to find them:
Divide both sides by 60:
x y z
1
5 4 3
Its now easy to see that the following points are on the plane:
5, 0, 0 0, 4, 0 0, 0,3
Now wheres the graph???
Inequalities in 3D
Inequalities in 3D
Inequalities in 3D
Inequalities in 3D
Inequalities in 3D
Inequalities in 3D
Inequalities in 3D
