1
If , then the limit exists if and only if:
A. exists.
B. and exist.
C. , , and all exist.
D. The magnitude exists.
Correct Answer: , , and all exist.
Explanation: The limit of a vector function exists if and only if the limits of its individual component functions exist.
2
A vector function is continuous at if:
A.
B.
C. is defined.
D. The derivative exists.
Correct Answer:
Explanation: Continuity requires that the limit as approaches equals the function value at .
3
Which of the following represents the derivative of the dot product ?
Correct Answer:
Explanation: The derivative of a dot product follows the product rule: .
4
If a vector function has a constant magnitude , then:
Correct Answer:
Explanation: Since , differentiating gives , implying and are orthogonal.
5
The length of a space curve defined by from to is given by:
Correct Answer:
Explanation: Arc length .
6
If , what curve does this represent?
A. Circle
B. Ellipse
C. Circular Helix
D. Parabola
Correct Answer: Circular Helix
Explanation: The and components form a circle of radius , while the component increases linearly with , forming a helix.
7
The unit tangent vector is defined as:
Correct Answer:
Explanation: The unit tangent vector is the velocity vector divided by its magnitude (speed).
8
For a particle moving along a curve , the velocity vector is:
A. Parallel to the curve.
B. Tangent to the path of motion.
C. Normal to the path of motion.
D. Independent of the path.
Correct Answer: Tangent to the path of motion.
Explanation: The velocity vector is always tangent to the curve traced by the particle.
9
If is the position vector, the acceleration vector is given by:
Correct Answer:
Explanation: Acceleration is the rate of change of velocity, which is the second derivative of position.
10
The speed of a particle with velocity is:
Correct Answer:
Explanation: Speed is the magnitude of the velocity vector.
11
The tangential component of acceleration is given by:
Correct Answer:
Explanation: .
12
The normal component of acceleration is given by:
Correct Answer:
Explanation: is related to curvature and is given by .
13
The Del operator () is defined as:
Correct Answer:
Explanation: The Del operator is a vector differential operator involving partial derivatives.
14
If is a scalar function, then is called the:
A. Divergence of
B. Curl of
C. Gradient of
D. Laplacian of
Correct Answer: Gradient of
Explanation: When acts on a scalar field , the result is the Gradient of , a vector field.
15
The gradient of a scalar field represents a vector that is:
A. Tangent to the level surface .
B. Normal to the level surface .
C. Parallel to the -axis.
D. Always zero.
Correct Answer: Normal to the level surface .
Explanation: Geometrically, the gradient vector at a point is normal (perpendicular) to the level surface passing through that point.
16
If and , then equals:
Correct Answer:
Explanation: .
17
Calculate .
Correct Answer:
Explanation: . Thus, .
18
The directional derivative of in the direction of a vector is given by:
Correct Answer:
Explanation: The directional derivative is the component of the gradient in the direction of the unit vector .
19
The directional derivative of is maximum in the direction of:
A. The position vector
B. The gradient
C. Any unit vector
D. The tangent vector
Correct Answer: The gradient
Explanation: The dot product is maximum when , i.e., in the direction of .
20
The maximum value of the directional derivative of at a point is:
Correct Answer:
Explanation: The maximum rate of change occurs in the direction of the gradient and its magnitude is .
21
The unit normal vector to the surface at a point is:
Correct Answer:
Explanation: Since is the normal vector, dividing it by its magnitude gives the unit normal vector.
22
If , the divergence is:
A.
B.
C.
D. A vector field.
Correct Answer:
Explanation: Divergence is the dot product of the Del operator and the vector field, resulting in a scalar sum of partial derivatives.
23
The divergence of a vector field yields a:
A. Vector field
B. Scalar field
C. Tensor
D. Matrix
Correct Answer: Scalar field
Explanation: The dot product results in a scalar quantity.
24
A vector field is called solenoidal if:
Correct Answer:
Explanation: A solenoidal field has zero divergence at all points (like an incompressible fluid).
25
Calculate the divergence of the position vector .
Correct Answer: 3
Explanation: .
26
The curl of a vector field , denoted , results in a:
A. Scalar field
B. Vector field
C. Constant
D. Zero
Correct Answer: Vector field
Explanation: The cross product of the Del operator and a vector field results in a vector field.
27
If , then the vector field is called:
A. Solenoidal
B. Irrotational
C. Divergent
D. Rotational
Correct Answer: Irrotational
Explanation: An irrotational field has zero curl, meaning there is no local rotation.
28
If is a conservative vector field, then there exists a scalar potential such that:
Correct Answer:
Explanation: A conservative field is the gradient of a scalar potential function.
29
Which of the following identities is always true for any smooth scalar field ?
Correct Answer:
Explanation: The curl of a gradient is always the zero vector.
30
Which of the following identities is always true for any smooth vector field ?
Correct Answer:
Explanation: The divergence of a curl is always zero.
31
The Laplacian operator is defined as:
Correct Answer:
Explanation: The Laplacian is the divergence of the gradient, i.e., .
32
Calculate , where .
Correct Answer:
Explanation: .
33
If where is a constant vector, then equals:
Correct Answer:
Explanation: The curl of the linear velocity of a rigid body rotation is twice the angular velocity vector.
34
If , find at .
Correct Answer:
Explanation: . At , this is .
35
The directional derivative of at in the direction of vector is:
Correct Answer: 2
Explanation: . At , . Direction is . Result: .
36
Which term describes the physical interpretation of Divergence?
A. Circulation per unit area
B. Net outward flux per unit volume
C. Rate of change in a specific direction
D. Maximum rate of increase
Correct Answer: Net outward flux per unit volume
Explanation: Divergence measures the extent to which a vector field acts as a source or sink at a given point.
37
Which term describes the physical interpretation of Curl?
A. Expansion of fluid
B. Flux across a surface
C. Rotation or circulation at a point
D. Scalar potential
Correct Answer: Rotation or circulation at a point
Explanation: Curl measures the infinitesimal rotation of a vector field in 3D space.
38
The derivative of is:
Correct Answer:
Explanation: The product rule for cross products preserves the order of the vectors.
39
If is a unit vector for all , then is:
A. 1
B.
C. -1
D. Undefined
Correct Answer:
Explanation: Since , differentiating gives .
40
Find .
Correct Answer:
Explanation: The divergence of the gradient is the Laplacian.
41
For a constant vector , is:
Correct Answer:
Explanation: Derivatives of constants are zero.
42
For a constant vector , is:
Correct Answer:
Explanation: Derivatives of constants are zero, so the curl is the zero vector.
43
The angle between the surfaces and at the point is determined by the angle between:
A. The position vectors at the point.
B. The gradient vectors of the surfaces at the point.
C. The tangent planes.
D. The -axis and the point.
Correct Answer: The gradient vectors of the surfaces at the point.
Explanation: The angle between surfaces is defined as the angle between their normal vectors (gradients) at the intersection point.
44
Which of the following is a scalar quantity?
A. Gradient
B. Curl
C. Divergence
D. Velocity
Correct Answer: Divergence
Explanation: Gradient and Curl produce vectors; Velocity is a vector; Divergence produces a scalar.
45
The necessary and sufficient condition for a vector function to be constant is:
Correct Answer:
Explanation: If the derivative is zero everywhere, the function does not change.
46
If , what is ?
Correct Answer:
Explanation: .
47
If is a scalar function and is a vector function, then is:
Correct Answer:
Explanation: This is the standard product rule for a scalar multiplying a vector.
48
The gradient of is:
Correct Answer:
Explanation: .
49
The vector field represents:
A. A divergent field
B. A rotational field (vortex)
C. A conservative field
D. A gradient field
Correct Answer: A rotational field (vortex)
Explanation: and . It represents rotation.
50
In the context of motion, curvature is related to the unit tangent and arc length by:
Correct Answer:
Explanation: Curvature is defined as the magnitude of the rate of change of the unit tangent vector with respect to arc length.