WHAT HAVE WE LEARNT SO FAR?• Classification of vector & scalar fields• Differential length, area and volume• Line, surface and volume integrals• Del operator• Gradient of a scalar• Divergence of a vector
– Divergence theorem• Curl of a vector
– Stokes’ theorem• Laplacian of a scalar
SCALAR AND VECTOR FIELD What is scalar field?
Quantities that can be completely described from its magnitude and phase. i.e. weight, distance, speed, voltage, impedance, current, energy
What is a vector field? Quantities that can be completely described
from its magnitude, phase and LOCATION. i.e. force, displacement, velocity, electric field, magnetic field
Need some sense of direction i.e. up, down right and left to specify
SCALAR AND VECTOR FIELD (CONT.) Is temperature a scalar quantity?A. YesB. NoAnswer: A, because it can be completely described by a number when someone ask how hot is today.
Is acceleration a scalar quantity?A. YesB. NoAnswer: B, because it requires both magnitude and some sense of direction to describe i.e. is it accelerating upward, downwards, left or right etc.
VECTOR CALCULUS What is vector calculus?
Concern with vector differentiation and line, surface and volume integral
So why do we need vector calculus?? To understand how the vector quantities i.e. electric field,
changes in space (vector differential) To determine the energy require for an object to travel
from one place to another through a complicated path under a field that could be spatially varying (line integral) i.e.
To pass ELEC 3600!! (vector differential and line integral)
W =
DIFFERENTIAL LENGTH, VOLUME AND SURFACE (CARTESIAN COORDINATE)
Differential length A vector whose magnitude is
close to zero i.e. dx, dy and dz → 0
Differential volume An object whose volume
approaches zero i.e. dv = dxdydz → 0 (scalar)
Differential surface A vector whose direction is
pointing normal to its surface area
Its surface area |dS| approach zero i.e. shaded area ~ 0
Calculated by cross product of two differential vector component
Differential is infinitely small difference between 2 quantities
DIFFERENTIAL LENGTH, VOLUME AND SURFACE (CYLINDRICAL
COORDINATE)All vector components MUST
have spatial units i.e. meters, cm, inch etc.
DIFFERENTIAL LENGTH, VOLUME AND SURFACE (SPHERICAL COORDINATE)
z
x
y
All vector components MUST have spatial units i.e. meters,
cm, inch etc.
LINE INTEGRAL Line integral: Integral of the tangential component
of vector field A along curve L. 2 vectors are involve inside the integral Result from line integral is a scalar
Line integral Definite integral
Diagram
Maths description
Result
Area under the curve
A measure of the total effect of a given field along a given pathInformation
required1. Vector field
expression A2. Path expression
1. Function f(x)2. Integral limits
Integral limits depends on path
SURFACE & VOLUME INTEGRAL Surface integral: Integral of the normal
component of vector field A along curve L. Two vectors involve inside the integral Result of surface integral is a scalar
Volume integral: Integral of a function f i.e. inside a given volume V. Two scalars involve inside the integral Result of volume integral is a scalar
SURFACE & VOLUME INTEGRAL (CONT.)Surface integral Volume integral
Diagram
Maths description
Result
A measure of the total effect of a scalar function i.e. temperature, inside a given volume
A measure of the total flux from vector field passing through a given surface
Information required
1. Vector field expression A
2. Surface expression
1. Scalar Function rv
2. Volume expression
Integral limits depends on surface
Integral limits depends on volume
PROBLEM 1 Given that , calculate
the circulation of F around the (closed) path shown in the following figure.
zyx2 aaaF 2yxzx
Solution:
DEL OPERATOR Vector differential operator Must operate on a quantity (i.e. function or
vector) to have a meaning
Mathematical form:Cartesian Cylindrical Spherical
SUMMARY OF GRAD, DIV & CURLGradient Divergence Curl
must operate on Scalar f(x,y) Vector A Vector A
Expression (Cartesian)
Expression (Cylindrical)
Expression (Spherical)
Result Vector Scalar Vector
SUMMARY OF GRAD, DIV & CURLGradient Divergence Curl
Physical meaning
A vector that gives direction of the maximum rate of change of a quantity i.e. temp
A scalar that measures the magnitude of a source or sink at a given point
Sink Source
A vector operator that describes the rotation/ununiformity of a vector field
RHC rotation
LHC rotation
Irrotational
i.e. Flux out < flux in
i.e. Flux out > flux in
IncompressibleFlux out = flux in
DIVERGENCE THEOREM Divergence theorem:
Total outward flux of a vector field A through a closed surface S is the same as the volume integral of divA. i.e. Transformation of volume integral involving divA to surface integral involving A
Equation:
Physical meaning: The total flux from field A passing through a volume V is equivalent to summing all the flux at the surface of V.
PROBLEM 2 (MIDTERM EXAM 2013)Verify the divergence theorem for the vector r2ar within the semisphere.
STOKE’S THEOREM Stoke’s Theorem:
The line integral of field A at the boundary of a closed surface S is the same as the total rotation of field A at the surface. i.e. Transformation of surface integral involving curlA to line integral of A
Equation:
Physical meaning: The total effect of field A along a closed path is equivalent to summing all the rotational component of the field inside the surface of which the path enclose.
LAPLACIAN OF A SCALAR FUNCTION
U is a scalar function of x, y, z (i.e. temperature)
Laplacian of a scalar = Divergence of a Gradient of scalar function.
Important operator when working with MAXWELL’S EQUATION!!
2 2 2
22 2 2
U U UU Ux y z
2 2 22
2 2 2, ,x y z
PROBLEM 3Given that , find(a) Where L is shown in the following figure(b) Where S is the area bounded by L(c) Is Stokes’s theorem satisfied?
yx aaF yyx 2
LdlF
S
dSF
1 2
3