Vector Calculus

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Vector Calculus
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Basic Vector Algebra

• Scalars are quantities having only a magnitude.
• Length, mass, temperature etc.

• Vectors are quantities having both a magnitude
  and a direction.
• Force, velocity, acceleration etc.

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Coordinate System
Rectangular or Cartesian Coordinate
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Coordinate System
Cylindrical Coordinate
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Coordinate System
Spherical Coordinate
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Vectors in Cartesian Coordinate System
A A1i A2j A3k (A1, A2, A3)
i, j, and k are unit vectors pointing in the
positive x, y, and z directions
A1, A2 and A3 are called x, y, and z component of
vector A
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Vectors in Cartesian Coordinate System
Magnitude of A
Direction of A
It is a unit directional vector !
Equality If A B, it means A B and also
Does the equality of two vectors necessarily
imply that they are identical?
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Addition of Vectors
Adding corresponding components
A (A1, A2, A3)
A B (A1 B1, A2 B2, A3 B3)
B (B1, B2, B3)
Geometrical representation
(a) A pair of vectors A and B
(b) Added by the head-to-tail method
(c) Added by parallelogram law
(d) B is subtracted from A
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Multiplication of a Vector by a Scalar
aA (aA1, aA2, aA3),
a is a real number
multiply the length of the vector by a the
direction unchanged
If a gt 0,
What happens if a lt 0?
multiply the length of the vector by a the
direction changed by 180
If a lt 0,
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Basic Properties of the Above Algebraic
1. Commutative law A B B A. 2. Associate
law (A B) C A (B C). 3. Zero vector
(0, 0, 0) A (0, 0, 0) A. 4. Negative
vector -A (-A1, -A2, -A3). 5. a(A B) aA
aB. 6. (aß)A a(ßA) 7. (a ß)A aA ßA.
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Dot Product
If A A1i A2j A3k and B B1i B2j B3k
It's Scalar, NOT Vector!
A . B A1B1 A2B2 A3B3
Another name scalar product.
Example If A (1, -3, 2) and B (4, 5, -8),
the dot product of A and B is -27.
Basic properties of the dot product

• A . B B . A
• (A B) . C A . C B . C
• A . (0, 0, 0) (0, 0, 0)
• A . A A2 A2

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Geometric Interpretation of Dot Product
A . B ABcosq ABcosq
If two nonzero vectors A . B 0, then
?
cosq 0
q 90
Perpendicular
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Cross Product
If A A1i A2j A3k and B B1i B2j B3k
Vector Product!
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Geometric Interpretation
If two nonzero vectors A B 0, then
?
sinq 0
q 0or 180
Parallel
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Example
Let A (1, -3, 2) and B (4, 5, -8), then
Basic properties of cross product

• A B -B A
• (A B) C A C B C
• A (B C) (A . C)B (A . B)C

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Vector and Scalar Functions
A vector valued function A(t) is a rule that
associates with each real number t a vector A(t).
A(t) A1(t)i A2(t)j A3(t)k
For example, f(t) t3 2t 4 is a scalar
function of a single variable t, while A(t) cos
ti sin tj tk is a vector function of t.
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Vector Differentiation
A vector function A(t) is differentiable at a
point t if
exists, and A'(t) is called the derivative of
A(t), written as
A'(t) A1'(t)i A2'(t)j A3'(t)k
Calculate the derivative of each component!
Example Let A(t) cos ti sin tj tk. Find
the derivative of A(t). Solution
A'(t) -sin ti cos tj k
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Rules of Vector Differentiation
if A constant.
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Vector Integration
Let A(t) A1(t)i A2(t)j A3(t)k and suppose
that the component functions A1(t), A2(t) and
A3(t) are integrable. Then the indefinite
integral of A(t) is defined by
Calculate the integral of each component!
If A1(t), A2(t) and A3(t) are integrable over the
interval t1, t2, then the definite integral of
A(t) is defined by
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Example
Let A(t) cos ti sin tj tk. Find
Solution
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Line Integral of Vector Functions
dl dxi dyj dzk
For a closed loop, i.e. ABCA,
circulation of P around L
Line given by L(x(s), y(s), z(s)), s parametric
variable
Always take the differential element dl as
positive and insert the integral limits according
to the paths!!!
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Example
For F yi xj, calculate the circulation of F
along the two paths as shown below.
Solution
dl dxi dyj dzk
Along path C2
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Example - Continue
Along path C1
Using x as the parametric variable, the path
equations are given as
Therefore,
and
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Example - Continue
The vector field defined by F in a given domain
is non-conservative. The line integral is
dependent on the integration path!
is work done on an object along path C if F
force !!
Is the static Electrical field conservative?
Yes, because the work done when we move a charge
from one point to another is independent of the
path but determined by the potential difference
between these two points.
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Surface Integral
Surface integral or the flux of P across the
surface S is
is the outward unit vector normal to the
surface.
For closed surface,
net outward flux of P.
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Example
If F xi yj (z2 1)k, calculate the flux of
F across the surface shown in the figure.
Solution
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Volume Integral
Evaluation choose a suitable integration order
and then find out the suitable lower and upper
limits for x, y and z respectively.
Example Let F 2xzi xj y2k. Evaluate
where V is the region bounded by the surface x
0, x 2, y 0, y 6, z 0, z 4.
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Volume Integral
Solution
In electromagnetic,
Total charge within the volume
where ?v volume charge density (C/m3)
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Scalar Field
Every point in a region of space is assigned a
scalar value obtained from a scalar function f(x,
y, z), then a scalar field f(x, y, z) is defined
in the region, such as the pressure in atmosphere
and mass density within the earth, etc.
Partial Derivatives
Mixed second partials
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Example
Let f x2 2y2. Calculate
and
Solution

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Gradient
Del operator
Gradient
Gradient characterizes maximum increase. If at a
point P the gradient of f is not the zero vector,
it represents the direction of maximum space rate
of increase in f at P.
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Example
Given potential function V x2y xy2 xz2, (a)
find the gradient of V, and (b) evaluate it at
(1, -1, 3).
Solution (a)
(b)
Direction of maximum increase
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Vector Field
Electric field E E(x, y, z),
Magnetic field H H(x, y, z)
Every point in a region of space is assigned a
vector value obtained from a vector function A(x,
y, z), then a vector field A(x, y, z) is defined
in the region.
R(t1, t2) acos t1i asin t1j t2k
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Divergence of a Vector Field
Representing field variations graphically by
directed field lines - flux lines
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Divergence of a Vector Field
The divergence of a vector field A at a point is
defined as the net outward flux of A per unit
volume as the volume about the point tends to
zero
It indicates the presence of a source (or sink)!
? term the source as flow source. And div A is a
measure of the strength of the flow source.
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Divergence of a Vector Field
In rectangular coordinate, the divergence of A
can be calculated as
For instance, if A 3xzi 2xyj yz2k, then
div A 3z 2x 2yz
At (1, 2, 2), div A 0 at (1, 1, 2), div A 4,
there is a source at (1, 3, 1), div A -1,
there is a sink.
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Curl of a Vector Field
The curl of a vector field A is a vector whose
magnitude is the maximum net circulation of A per
unit area as the area tends to zero and whose
direction is the normal direction of the area.
It is an indication of a vortex source, which
causes a circulation of a vector field around it.
Water whirling down a sink drain is an example of
a vortex sink causing a circulation of fluid
velocity.
If A is electric field intensity, then the
circulation will be an electromotive force around
the closed path.
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Curl of a Vector Field
In rectangular coordinate, curl A can be
calculated as
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Curl of a Vector Field
Example
If A yzi 3zxj zk, then