Unit5 - Subjective Questions

Limit of a Vector Function:
Let be a vector function of a scalar variable . Then a vector is said to be the limit of as if, for every , there exists a such that:

Symbolically, we write: .

Continuity of a Vector Function:
A vector function is said to be continuous at if:

1. is defined.
2. exists.
3. .

If , then is continuous at if and only if the scalar functions , and are continuous at .

Let be a vector function with constant magnitude.
Let , where is a constant.

We know that:

Differentiating both sides with respect to :

Using the product rule for dot products:

Since the dot product is commutative ():

Therefore:

Geometric Interpretation: This implies that for a vector of constant magnitude, the derivative vector (tangent) is always perpendicular to the vector itself.

Definition:
The length of a space curve is the limit of the lengths of inscribed polygons as the number of segments increases indefinitely and the length of the largest segment approaches zero.

Formula:
If a space curve is defined by the position vector where , and the derivatives are continuous, the length of the curve from to is given by:

Expanded in Cartesian coordinates:

Let the position of a particle moving on a curve be given by at time .

1. Velocity Vector ():
The velocity is the rate of change of the position vector with respect to time.

The magnitude is called the speed.
The direction of is along the tangent to the path of motion.

2. Acceleration Vector ():
The acceleration is the rate of change of the velocity vector with respect to time.

Acceleration represents how the velocity (speed and/or direction) changes over time.

Let be the position vector. Velocity is , where is speed and is the unit tangent vector.

Acceleration .
Using the product rule:

Using the Frenet-Serret formula , and chain rule :

Where is curvature and is the radius of curvature.

Tangential Component ():

Alternatively calculated as:

Normal Component ():

Alternatively calculated as:

Definition:
Let be a differentiable scalar point function. The gradient of , denoted by or , is defined as:

Note that is a vector quantity.

Physical/Geometrical Significance:

1. Normal Vector: represents a vector normal (perpendicular) to the level surface at a given point.
2. Rate of Change: The magnitude represents the maximum rate of change of the function . The direction of is the direction in which increases most rapidly.

Definition:
The directional derivative of a scalar point function at a point in the direction of a vector is the rate of change of at with respect to distance measured in the direction of .

Formula:
If is the unit vector in the direction of (i.e., ), then the directional derivative (D.D.) is given by:

Where .

Note: The directional derivative is maximum in the direction of the gradient, and the maximum value is .

Let the surface be defined by the scalar function .

The normal vector to the surface is given by the gradient .

Partial Derivatives:

At point :

So, .

Unit Normal Vector ():

Divergence:
Let be a continuously differentiable vector point function. The divergence of , denoted by or , is a scalar defined as:

Physical Meaning: It represents the net rate of flux (outflow minus inflow) of the vector field per unit volume. Positive divergence implies a source, negative implies a sink.

Solenoidal Vector:
A vector field is said to be Solenoidal if its divergence is zero at all points.

This implies physically that the fluid (or field) is incompressible; there is no net generation or loss of fluid in the region.

Curl:
Let be a differentiable vector field. The Curl of , denoted by or , is a vector defined by the determinant:

Physical Meaning: Curl represents the angular velocity or rotation of the vector field at a point. It measures the "swirl" of the field.

Irrotational Vector:
A vector field is said to be Irrotational if its curl is the zero vector.

If a field is irrotational, it is a conservative field, meaning for some scalar potential .

We need to prove that .

Let be a scalar function.
.

Now, calculate the curl of this vector:

Expanding the determinant:

Assuming has continuous second-order partial derivatives, the order of differentiation does not matter (Clairaut's Theorem):
, etc.

Therefore:

Thus, the curl of a gradient is always zero.

1. Check for Solenoidal ():
Let .

Since divergence is 0, is Solenoidal.

2. Check for Irrotational ():

-comp:
-comp:
-comp:

Since , is Irrotational.

We know and .
Differentiating partially w.r.t : .
Similarly, and .

Now, .

Using chain rule:
.

Substituting this back:


.

Hence proved.

The angle between two surfaces at a point is the angle between their normal vectors (gradients) at that point.

Surface 1:

At : .

Surface 2:

At : .

Calculate Angle :

Dot Product: .
Magnitude .
Magnitude .

If a vector field is conservative (irrotational), there exists a scalar potential such that .
To find :

1. Check Condition: Verify .
2. Set Equations: Write as .
   This gives a system of partial differential equations:
   
3. Integrate:
   • Integrate with respect to (treating constant) adding an arbitrary function .
   • Integrate with respect to (treating constant).
   • Integrate with respect to (treating constant).
4. Combine: Compare the integrals to determine the unknown functions and combine terms to form the single function .

Exact Differential Method:
Alternatively, write . Group terms to form exact differentials and integrate.

Given position vector .

1. Divergence:

.

2. Curl:

Since are independent variables:
.

Result: and .

We use the tensor/summation notation or determinant expansion properties.
is complex, so let's use the operator property (where subscript denotes which function the operator acts upon).

Using scalar triple product property and cyclic permutations:

1st part: .
Summing over axes gives .

2nd part: .
Summing over axes gives .

Combining both results:
.

The Laplacian Operator, denoted by (del squared) or , is a second-order differential operator.

It is defined as the divergence of the gradient of a scalar function .

Expression:

In Cartesian coordinates :
If is a scalar field:

Taking the divergence:

1. Find Gradient ():

2. Evaluate at :

3. Find Unit Direction Vector ():
Vector .
.
.

4. Calculate Directional Derivative:
D.D.

.

Result: .

For to be irrotational, .

Where , , .

Components of Curl:

1. component:
   .

2. component:
   .

3. component:
   .

Result: .