Vector Analysis

Summary of prerequisites (revision)

  1. A scalar quantity has magnitude only; a vector quantity has both magnitude and direction.
  1. The axes of reference, OX, OY, OZ, form a right-handed set. The symbols i, j, k denote unit vectors in the directions of OX, OY, OZ, respectively.

If then

  1. The direction cosines are the cosines of the angles between the vector r and the axes OX, OY, OZ, respectively. For any vector r = axi+ayj+azk

  1. Scalar Product (Dot Product)

where is the angle between A and B.

If A = axi+ayj+azk and B = bxi+byj+bzk then

  1. Vector product (Cross Product)

A x B = C =ABsinθn ; n is a unit vector in a direction perpendicular to A and B

  1. Angle between two vectors

For perpendicular vectors

For parallel vectors

Solve!:

a)If P is the point (3,2,6), determine r, , l, m, and n

b)If A = 2i+3j+4k and B = i2j+3k, find the direction cosines of each vector hence the angle between A and B

c)If A = 2i3j+4k and B = i+2j+5k, determineand verify

d)If A = 3i2j+4k and B = 2i3j5k, determineand verify

Answers:

a)

b)

c)

d)

Triple Product of three vectors

If

Then

Remembering that

We have

Example: If

Properties of scalar triple product

a)

Since interchanging two rows in a determinant reverses the sign.

It can be shown

i.e the scalar triple product is unchanged by a cyclic change of the vectors involved.

b)

i.e a change of vectors not in cyclic order, changes the sign of the scalar triple product

c) since two rows are identical

Coplanar vectors

Vector triple products of three vectors:

If

Take note that

It can be proved that:

Example: If

Equivalently using

Try these: Determine for

a)

Answer:

b)

Answer:

Differentiation of vectors

Many practical problems deal with vectors that change with time (independent scalar) e.g. velocity, acceleration etc. If A can be represented as

Differentiating with respect to t

In general if u is the independent scalar parameter

Example: A particle moves in space so that at time t its position is stated by:

Find the components of its velocity and acceleration in the direction of the vector when t = 1

At t = 1

A unit vector parallel to

The component of (velocity) in the direction of :

The component of (acceleration) in the direction of :

Differentiation of sums and products of vectors

If and then

a)

b)

c)

d)

Proof that and are perpendicular vectors.

If , then

But

If and 

Unit tangent vectors

If a position vector in space, the direction of vector is parallel to the tangent to the curve at P.

 the unit tangent vector

Example: Determine the unit tangent vector at point (2,4,7) for the curve with parametric equations

The point (2,4,7) corresponds to u = 1

The vector equation of the curve is:

Integration of vector functions

If where F, x, y, z are expressed as function of u

Example: If then

Example: If . Evaluate

Try yourself! : Determine where and

Ans:

Scalar and vector fields

Grad (gradient of a scalar function) denoted by

If a scalar function (x,y,z), is continuously differentiable with respect to its variables x, y, z, throughout the region, the gradient of , written as or , is defined as the vector:

=

Where is called a vector differential operator and is denoted by  (pronounced ‘del’ or sometimes ‘nabla’)

Example: If , determine at the point P(1,3,2)

Try these: If and , determine

Ans:

Consider these:

Directional derivatives

is the projection of on the unit vector and is called the directional derivative of in the direction of . It gives the rates of change of with distance measured in the direction of .

will be a maximum when and have the same direction.

Since as when and are parallel.

Example: Find the directional derivative of the function at the point (1,2,1) in the direction of the vector

Try these:

a) Find the directional derivative ofat the point (1,1,2) in the direction of the vector

Ans:

b) Find the direction from the point (1,1,0) which gives the greatest rate of increase of the function

Ans:

Unit normal vectors

If , this relationship represents a surface in space, depending on the value ascribed to the constant.

is a vector perpendicular to the surface at P, in the direction of maximum rate of change of . The magnitude of the maximum rate of change is given by

Example: Find the unit normal vector to the surface at the point (1,3,1)

At (1,3,1)

The unit normal at (1,3,1) =

Collecting our results so far, we have, for a scalar function :

a)

b)

c) directional derivative

d) unit normal vector

Grad of sum and products of scalars

a)

b)

Div (divergence of a vector function) denoted by

The operator can be applied to a vector function to give the divergence of . If

Try these: Find for:

a)

b)

A vector for which at all points is called solenoid vector.

Curl (curl of a vector function) denoted by

.The curl operator , acts on vector and gives another vector.

If

Try these: Find for:

a) at the point (1,3,2) Ans:

b) at the point (2,0,3) Ans:

Summary of grad, div and curl

a)Grad operator acts on a scalar field to give a vector field (vector differential operator)

b)Div operator acts on a vector field to give a scalar field

c)Curl operator acts on a vector field to give a vector field

d)With scalar function

e)With a vector function

i)

ii)

Revision exercise! If and determine for point P(1,1,2):

a) Ans:

b) unit normalAns:

c) Ans:

d) Ans:

Multiple operations

Example: If . Find

Try these:

a) If determine at the point (2,4,1)

Ans:

b) If determine at the point (2,1,1)

Ans:

Remember that grad, div and curl are operators that they must act on a scalar or vector as appropriate. They cannot exist alone.

Some interesting results but you have to prove it!

a)

b)

c)