13.1_13.2_MATB_113_Notes_E.doc

16 PARTIAL DIFFERENTIATION

13. VECTOR-VALUED FUNCTIONS

AND MOTION IN SPACE 13.1 Curves in Space and Their TangentsA vector-valued function r is represented by

where and are scalar functions.

Illustration on vector valued function Example:1 Graphing a vector valued function

(i) Sketch .

(ii) Sketch .Limits and Continuity

Definitions Limit of Vector functions

Let be a vector function and L a vector, we say that r has limit L as t approaches to and write

if, for every number , there is a corresponding number such that for all t

Example:2

Find the .

Definition: Continuous at a PointA vector function is continuous at a point in its domain if

.The function is continuous if it is continuous at every pointing its domain.

Example:3

Show that the function is discontinuous at a certain point t.

Derivatives and Motion

Definition Derivative

The vector function has a derivative (is differentiable) at t if

f, g and h have derivatives at t. The derivative is the vector function

A curve that is made up of a finite number of smooth curves pieced together in a continuous fashion is called piecewise smooth.

Definitions: Velocity, direction, Speed,

Acceleration

If r is the position vector of a particular moving along a smooth curve in space, then

is the particles velocity vector, tangent to the curve. At any time t, the direction of

v is the direction of motion, the magnitude of v is the particles speed, and the derivative , when it exists, is the particles acceleration vector. In summary

1. Velocity is the derivative of position:

2. Speed is the magnitude of the velocity:

Speed =

3. Acceleration is the derivative of

velocity:

4. The unit vector is the direction of

motion at time t.

Example:4

The position of a particle in space at time t is given by

.

Find (a) the velocity and acceleration vector at t = 2.

(b) the speed at any time t.

(c) the time t, if any when the particles acceleration is

orthogonal to its velocity.

Differentiation Rules for Vector Functions

Let u and v be differentiable vector functions of t, C a constant vector, c any scalar, and f any differentiable scalar function.

1. Constant function Rule:

2. Scalar multiple Rule:

3. Sum rule:

4. Difference Rule:

5. Dot Product Rule:

6. Cross Product Rule:

7. Chain Rule:

Proof of the Dot Product Rule

Proof of the Chain Rule

Vector Functions of Constant Length

If r is a differentiable vector function of t of constant

Length, then

Example:5

Show that has constant length and is orthogonal to its derivative.

13.2 Integrals of Vector Functions; Projectile

MotionDefinition Indefinite Integral

The indefinite integral of r with respect to t is the set of all antiderivatives of r denoted by . If R is any antiderivative of r, then

Example:6

Evaluate .Definition Definite Integral

If the components of

are integrals over ,

then so is r, and the definite integral of r from a to b is

Example: 7

Evaluate .Example: 8

Find the vector valued function r(t) given by

with initial conditions and

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