Unit6 - Subjective Questions
MTH166 • Practice Questions with Detailed Answers
Define the Line Integral of a vector function and explain the concept of Work Done by a force field.
Line Integral:
Let be a continuous vector point function defined in a region containing a curve . Let be the position vector of a point on . The integral defined by defined by is called the line integral of along the curve .
Work Done:
If represents a force acting on a particle moving along an arc , then the total work done by the force is given by the line integral:
If and , then:
Evaluate the line integral where along the curve in the -plane given by from to .
Solution:
Given and .
The curve is , so . The limits for are from $0$ to $1$.
Substituting and into the integral:
Integrating with respect to :
Answer:
State Green's Theorem in a plane.
Statement of Green's Theorem:
If is a closed region in the -plane bounded by a simple closed curve , and if and are continuous functions having continuous partial derivatives and in , then:
Where the line integral over is taken in the positive (counter-clockwise) direction.
Verify Green's Theorem for where is the square with vertices .
Verification:
Here and .
1. RHS (Double Integral):
2. LHS (Line Integral):
Calculated along 4 sides of the square:
Sum .
Since LHS = RHS = 8, Green's Theorem is verified.
Using Green's Theorem, find the area of the region bounded by the ellipse .
Using Green's Theorem, the area of a region bounded by is given by:
Parametric equations for the ellipse are , , where .
Then, and .
Substituting these values:
Now, integrate:
Answer: Area =
Define Solenoidal and Irrotational vectors. How are they related to vector potentials?
Solenoidal Vector:
A vector function is said to be solenoidal if its divergence is zero, i.e., . This implies there is no net flux generated at any point.
Irrotational Vector:
A vector function is said to be irrotational if its curl is zero, i.e., . This implies the field is conservative.
Relation to Potentials:
- Scalar Potential: If is irrotational (), then can be expressed as the gradient of a scalar potential function , such that .
- Vector Potential: If is solenoidal (), then can be expressed as the curl of a vector potential , such that .
Determine if the vector field is conservative. If so, find the scalar potential.
Step 1: Check if Conservative (Irrotational)
Compute Curl :
- -component:
- -component:
- -component:
Since , is conservative.
Step 2: Find Scalar Potential
We need .
- . Differentiating (1) w.r.t : . Thus .
Update . - . Differentiating updated w.r.t : . Thus .
Final Potential:
State Stokes' Theorem.
Stokes' Theorem:
Let be an open surface bounded by a simple closed curve . If is a vector point function having continuous first order partial derivatives on , then:
Where:
- The line integral is taken around in the direction determined by the right-hand rule relative to the normal vector .
- is the unit outward normal vector to the surface .
- is the element of surface area.
Verify Stokes' Theorem for where is the upper half of the sphere and is its boundary.
Verification:
The boundary is the circle in the plane.
1. LHS: Line Integral over ()
on becomes .
Let . .
2. RHS: Surface Integral
Compute .
For the surface, (on unit sphere).
Using projection on -plane, .
This is the area of the unit circle , which is .
Conclusion: LHS = RHS = . Verified.
Using Stokes' theorem, evaluate where and is the curve of intersection of and .
By Stokes' Theorem: .
1. Calculate Curl:
.
2. Normal Vector:
The surface lies on the plane . The normal to the plane is .
Unit normal .
3. Dot Product:
.
4. Surface Integral:
is the area of the circle cut by the plane through the sphere. The intersection is a circle of radius . Since it passes through the center? No, the plane passes through and . The distance from origin to plane is .
Radius of circle .
Area of circle .
Wait, here is the surface area of the planar region enclosed by . The area is indeed .
State Gauss's Divergence Theorem.
Gauss's Divergence Theorem:
If is the volume bounded by a closed surface , and if is a vector point function with continuous partial derivatives in , then:
Where:
- is the unit outward normal vector to the surface .
- is the divergence of .
- is the element of volume.
Use Divergence Theorem to evaluate , where and is the surface bounding the region and .
Using Divergence Theorem: .
1. Calculate Divergence:
2. Setup Volume Integral:
The region is a cylinder with .
Convert to cylindrical coordinates: .
Limits: , , . .
3. Integrate w.r.t z:
4. Integrate w.r.t r:
5. Integrate w.r.t :
Answer:
Evaluate the surface integral where is the surface of the sphere in the first octant.
Here .
The surface is . Let .
Normal vector .
.
We need to evaluate over the first octant.
Using spherical coordinates:
.
(for unit sphere).
Limits: , .
integrand =
Integral
Part 1 (): Let , . .
Part 2 (): ? No. . Wait. . From 0 to 1 gives .
.
Answer:
Prove that , where is the volume enclosed by the closed surface and is the position vector.
Let be the position vector.
We are asked to evaluate .
Apply Gauss's Divergence Theorem:
Here, let .
Calculate the divergence of :
Substitute back into the volume integral:
Since (the volume of the region),
Verify Gauss's Divergence Theorem for taken over the rectangular parallelopiped .
1. Volume Integral (RHS):
Solving this yields .
2. Surface Integral (LHS):
The surface has 6 faces.
- Face : . . .
- Face : . . .
- Sum for x-faces: .
- Similarly, sum for y-faces: .
- Sum for z-faces: .
Total Surface Integral = .
Since LHS = RHS, the theorem is verified.
Distinguish between Line Integral, Surface Integral, and Volume Integral.
-
Line Integral:
- Integration domain: A curve (1D) in space.
- Notation: or .
- Physical meaning: Work done by a force along a path, circulation of fluid.
-
Surface Integral:
- Integration domain: A surface (2D) in space.
- Notation: .
- Physical meaning: Flux of a vector field across a surface (e.g., fluid flow rate, electric flux).
-
Volume Integral:
- Integration domain: A volume (3D) in space.
- Notation: or .
- Physical meaning: Total mass (from density), total charge, or total source strength in a region.
Find the work done in moving a particle once around a circle in the -plane, if the circle has center at the origin and radius $3$, and the force field is given by .
We can use Green's Theorem (plane) or Stokes' Theorem.
Since is in the -plane, .
becomes .
However, .
.
Apply Green's Theorem:
.
.
Area = .
Answer:
Apply Green's Theorem to evaluate where is the triangle bounded by .
Let .
.
.
Using Green's Theorem:
Region limits: goes from $0$ to . For a fixed , goes from $0$ to .
- .
- .
Explain the physical interpretation of the divergence of a vector field.
Physical Interpretation of Divergence:
The divergence of a vector field at a point , denoted by or , represents the rate of flux generation per unit volume at that point.
- Source: If at a point, the point acts as a source. There is a net outflow of the vector field from a small region surrounding the point (e.g., gas expanding, or positive electric charge).
- Sink: If , the point acts as a sink. There is a net inflow into the region (e.g., gas compressing, or negative electric charge).
- Incompressible/Solenoidal: If , there is no net gain or loss; the amount of flux entering equals the amount leaving (e.g., flow of an incompressible fluid).
If , prove that the line integral is independent of the path.
Let . The line integral along a curve from point to point is:
We know that .
Substituting this:
The result depends only on the potential function values at the end points and , and not on the path taken between them. Thus, the field is conservative and the line integral is independent of path.