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Vector Analysis II
Line integrals
Scalar field
Example: If , evaluate along the curve c having parametric equations between and
Try this: If evaluate along the curve c defined by between and .
Ans:
Vector field
i.e the line integral of from A to B along the stated curve =
Since and
Example: If , evaluate between and along the curve having parametric equation
If the vector field is a force field, then the line integral represents the work done in moving a unit particle along the prescribed curve c from A to B.
Example: If evaluate between and
a)along the straight lines c1 from then along the straight lines c2 from then along the straight lines c3 from
b)along the straight lines c4 from
Summing the three partial results
b)
c)Try yourself! along the straight lines c5 from then along the straight lines c6 from
The value of the line integral depends on the path taken between the two end point A and B
Try this: If evaluate along the curve between
Ans: 23.8
Surface integrals
The vector product of two vectors A and B is defined by:
Scalar fields
Example: A scalar field exists over the curves surface S defined by between the planes and in the first octant. Evaluate over this surface.
Given:
We have to evaluate the integral over the prescribed surface.
Changing to cylindrical coordinates with
Example:
Projection of dS on the xy-plane, gives
Vector fields
Example:
A vector field exists over a surface S defined by bounded by in the first octant.
Converting to polar coordinate
Conservative vector fields
If, however , the line integral between A and B is independent of the path of integration between the two end points, then the vector field F is said to be conservative.
It follows that, for a closed path in a conservative field
Hence, for the closed path
This result hold good only for a closed curve and when the vector field is a conservative field.
Example: If , evaluate the line integral between and
a)along the curve c whose parametric equations are
b)along the three straight lines ; ;
Hence determine whether or not F is a conservative field.
b)
i.e F is a conservative field
Further test on conservative field
If is a conservative field
a)
b) can be expressed as where is a scalar field to be determined
Example: We have shown that is a conservative field
a)
b) Express as where is a scalar in x,y,z
If
These three equations are satisfied if
where
Three test can be applied to determine whether or not a vector field is conservative:
a)
b)
c)
Exercise: Determine which of the following vector fields are conservative
a) b)
c) d)
Ans: a) NO, b) YES, c) YES, d) NO
Divergence theorem (Gauss’ theorem)
For a closed surface S, enclosing a region V in a vector field
In general, this means that the volume integral (triple integral) on the left hand side can be expressed as a surface integral (double integral) on the right hand side.
Example: Verify the divergence theorem for the vector field taken over the region bounded by the planes
a)To find
b)To find i.e
i)S1 (base): (outwards and downwards)
ii)S2 (top):
iii) S3 (right hand end):
iv) S4 (left hand end):
Stokes’ theorem
we can express a surface integral in terms of a line integral round the boundary curve
Example: A hemisphere S is defined by . A vector field exist over the surface and around its boundary c. Verify Stokes’ theorem that
a)
Converting to polar coordinates
b)
Expressing in spherical polar coordinates:
Direction normal to a surface S
When dealing with divergence theorem, the normal vectors were drawn in a direction outward from the enclosed region.
With an open surface, there is in fact no inward or outward direction. With any general surface, a normal vector can be drawn in either of two opposite directions. To avoid confusion, a convention agreed is as follows:
A unit normal is drawn perpendicular to the surface S at any point in the direction indicated by applying a right-handed screw sense to the direction of integration round the boundary c.
Example: A surface consist of five sections formed by the planes x=0, x=1, y=0, y=3, z=2 in the first octant. If the vector field exists over the surface and around its boundary, verify Stokes’ theorem
We have to verify that
a) Finding
i) along c1: y=0,z=0, dy=0, dz=0
ii) along c2: x=1; z=0; dx=0; dz=0
iii) along c3: y=3; z=0; dy=0; dz=0
iv) along c4: x=0; z=0; dx=0; dz=0
b) Finding
i) S1(top) :
ii) S2 (right-hand end):
iii) S3 (left-hand end):
Do the rest yourself!
Rustam Puteh