Directional Derivatives and Harmonic Functions - Question Bank

1. What is the gradient of the function f(x, y) = x²y at the point (1, 2)?
A) (4, 1)
B) (2, 1)
C) (4, 4)
D) (2, 4)
2. Which of the following is a consequence of the Maximum Modulus Principle for harmonic functions (related to the Maximum/Minimum Principle)?
A) A harmonic function attains its maximum value on the boundary of its domain.
B) A harmonic function attains its minimum value on the boundary of its domain.
C) A non-constant harmonic function cannot attain its maximum or minimum value inside the domain.
D) A harmonic function is always zero at the center of any sphere.
3. If the directional derivative of f at P in direction u is positive, it means f is increasing at P as you move in direction u.
A) True
B) False
C)
D)
4. The function f(x, y) = x² - y² is harmonic in the entire xy-plane.
A) True
B) False
C)
D)
5. Consider the function f(x, y) = cos(x)sinh(y). What is its Laplacian?
A) 0
B) cos(x)sinh(y)
C) -cos(x)sinh(y)
D) sin(x)cosh(y)
6. If f is harmonic and u is a unit vector, then the directional derivative of f in the direction u is:
A) The directional derivative of f in the direction -u.
B) The negative of the directional derivative of f in the direction -u.
C) Equal to the directional derivative of f in the direction of Vf.
D) Zero.
7. The directional derivative of f at P in the direction of u is the component of the gradient of f in the direction of u.
A) True
B) False
C)
D)
8. Which of the following is an example of a harmonic function in 3D?
A) f(x, y, z) = x² + y² + z²
B) f(x, y, z) = x + y + z
C) f(x, y, z) = 1/√(x² + y² + z²)
D) f(x, y, z) = x² + y² - 2z²
9. If f is harmonic, then the integral of Vf ⋅ n ds over a closed curve C is:
A) Zero
B) 2π
C) 1
D) ∫∫_R ∇²f dA
10. The directional derivative of f(x, y) = x²y³ at the point (1, 2) in the direction of the vector (1, -1) is:
A) -40/√2
B) 40/√2
C) -40
D) 40
11. Which of the following is a necessary condition for a function to be harmonic?
A) Its gradient must be zero.
B) Its divergence must be zero.
C) Its curl must be zero.
D) Its Laplacian must be zero.
12. If f is harmonic, then the integral of f over any closed curve C in its domain is:
A) Always positive
B) Always negative
C) Zero
D) Equal to 2π
13. What is the directional derivative of f(x, y) = √(x² + y²) at (3, 4) in the direction of the vector (0, 1)?
A) 3/5
B) 4/5
C) 1
D) 0
14. The set of all harmonic functions on a domain forms a:
A) Group under addition
B) Vector space
C) Field
D) Lattice
15. Consider the function f(x, y) = x³ - 3xy². What is the directional derivative of f at (1, 1) in the direction of the positive x-axis?
A) 0
B) 3
C) -3
D) 6
16. If a function is harmonic, its second partial derivatives are:
A) Always zero
B) Continuous
C) Discontinuous
D) Imaginary
17. Which theorem states that if f is harmonic in a domain D, then for any point P in D and any sphere centered at P lying entirely within D, the average value of f on the sphere is f(P)?
A) Gauss's Divergence Theorem
B) Stokes' Theorem
C) Mean Value Property
D) Maximum Modulus Principle
18. What is the directional derivative of f(x, y) = x² - y² at (2, 1) in the direction of the vector (3, 4)?
A) 4
B) 8
C) 12
D) 16
19. The function f(x, y) = e^x sin(y) is:
A) Harmonic
B) Not harmonic
C) Harmonic only if y=0
D) Harmonic only if x=0
20. If f(x, y) = ax + by + c, then f is:
A) Harmonic only if a=b=0
B) Harmonic for any a, b, c
C) Never harmonic
D) Harmonic only if c=0
21. What is the directional derivative of f(x, y) = 5 at the point (3, 4) in the direction of (1, 1)?
A) 0
B) 5
C) sqrt(2)
D) 5/sqrt(2)
22. Which physical phenomenon is often described by Laplace's equation?
A) Wave propagation
B) Diffusion of heat in steady state
C) Newton's law of cooling
D) Projectile motion
23. If Vf(P) = (0, 0), then the directional derivative of f at P in any direction u is:
A) ||u||
B) 1
C) 0
D) Undefined
24. The directional derivative of a constant function is always:
A) Positive
B) Negative
C) Zero
D) Undefined
25. What is the direction of the maximum rate of increase for f(x, y) = x²y at the point (1, 2)?
A) (2, 1)
B) (4, 1)
C) (4, 4)
D) (2, 4)
26. Consider f(x, y) = x³ - 3xy². Is f harmonic?
A) Yes
B) No
C)
D)
27. A function f is harmonic if and only if it is the real or imaginary part of an analytic function (in 2D).
A) True
B) False
C)
D)
28. If Vf = (2x, 3y) at point (1, 2), what is the directional derivative of f in the direction u = (1/√2, 1/√2)?
A) 5/√2
B) 7/√2
C) 5
D) 7
29. The directional derivative of f at P in the direction of u is the rate of change of f as one moves from P in the direction of u. This is a local property.
A) True
B) False
C)
D)
30. Which of the following is NOT a common application of harmonic functions?
A) Electrostatics
B) Fluid dynamics
C) Heat conduction
D) Quantum mechanics wave functions
31. If f(x, y, z) is a harmonic function, then the divergence of its gradient is:
A) ∇f
B) ∇ ⋅ ∇f
C) ∇ x ∇f
D) f
32. What is the Laplacian of the function f(x, y) = sin(x)cosh(y)?
A) 0
B) sin(x)cosh(y)
C) -sin(x)cosh(y)
D) cos(x)sinh(y)
33. The property that a harmonic function has no local maxima or minima in its domain (unless it is constant) is known as the:
A) Mean Value Property
B) Maximum Modulus Principle
C) Minimum Modulus Principle
D) Maximum/Minimum Principle
34. A function that is harmonic in a simply connected domain and has no singularities is called:
A) Analytic
B) Holomorphic
C) Harmonic polynomial
D) Harmonic function
35. Consider the function f(x, y) = ln(x² + y²). Is this function harmonic? (Assume x² + y² ≠ 0)
A) Yes, its Laplacian is 0.
B) No, its Laplacian is 2/(x² + y²).
C) No, its Laplacian is -2/(x² + y²).
D) Yes, its Laplacian is 1.
36. The Mean Value Property for harmonic functions states that the average value of a harmonic function over a sphere is equal to:
A) The value of the function at the center of the sphere.
B) Zero.
C) The maximum value of the function on the sphere.
D) The minimum value of the function on the sphere.
37. The Cauchy-Riemann equations are related to harmonic functions in complex analysis. If f(z) = u(x, y) + iv(x, y) is analytic, then u and v are:
A) Neither necessarily harmonic
B) Both harmonic
C) Only u is harmonic
D) Only v is harmonic
38. If f is a harmonic function, then its Laplacian is:
A) Always positive
B) Always negative
C) Zero
D) A constant
39. Which of the following functions is harmonic?
A) f(x, y) = x² + y²
B) f(x, y) = xy
C) f(x, y) = x² - y²
D) f(x, y) = e^(x+y)
40. The operator ∇² is known as the:
A) Gradient operator
B) Divergence operator
C) Laplacian operator
D) Curl operator
41. Laplace's equation in three dimensions is given by:
A) ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z² = 0
B) ∇f = ∂f/∂x + ∂f/∂y + ∂f/∂z = 0
C) ∇²f = ∂²f/∂x² - ∂²f/∂y² - ∂²f/∂z² = 0
D) ∇f = 0
42. Laplace's equation in two dimensions is given by:
A) ∂²f/∂x² + ∂²f/∂y² = 0
B) ∂f/∂x + ∂f/∂y = 0
C) ∂²f/∂x² - ∂²f/∂y² = 0
D) ∂f/∂x * ∂f/∂y = 0
43. What is a harmonic function?
A) A function whose gradient is zero everywhere.
B) A function that satisfies Laplace's equation.
C) A function whose Laplacian is a constant.
D) A function whose directional derivative is always positive.
44. Which of the following is NOT a property of the directional derivative?
A) It is linear in the direction vector.
B) It is equal to the gradient if the direction vector is a unit vector.
C) It is always non-negative.
D) It measures the rate of change along a specific direction.
45. For a function f(x, y, z), the directional derivative in the direction of vector v (not necessarily unit) is given by:
A) Vf ⋅ (v / ||v||)
B) Vf ⋅ v
C) (Vf ⋅ v) / ||Vf||
D) (Vf ⋅ v) / ||v||
46. Consider the function f(x, y) = x^2 + y^2. What is the directional derivative of f at (1, 1) in the direction of the vector (1, 0)?
A) 0
B) 2
C) 4
D) sqrt(8)
47. The directional derivative of f at P in the direction of u is zero if u is perpendicular to:
A) The Hessian matrix of f at P.
B) The gradient of f at P.
C) The Laplacian of f at P.
D) The divergence of f at P.
48. To find the maximum rate of increase of a function f at a point P, one should move in the direction of:
A) The negative gradient of f.
B) The vector tangent to the level curve of f at P.
C) The gradient of f at P.
D) The unit normal vector to the level surface of f at P.
49. The directional derivative measures the rate of change of a function:
A) Along a specific direction.
B) In all directions simultaneously.
C) Perpendicular to the direction of interest.
D) At the origin of the coordinate system.
50. If Vf represents the gradient of a scalar function f, and u is a unit vector, the directional derivative of f in the direction of u is given by:
A) u ⋅ (Vf)
B) u x (Vf)
C) Vf ⋅ Vf
D) Vf / ||Vf||
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