Directional Derivatives and Harmonic Functions - One Line Questions
1.
The directional derivative of f(x, y) = x²y³ at the point (1, 2) in the direction of the vector (1, -1) is: —
-40/√2
2.
What is the direction of the maximum rate of increase for f(x, y) = x²y at the point (1, 2)? —
(4, 1)
3.
What is the gradient of the function f(x, y) = x²y at the point (1, 2)? —
(4, 4)
4.
Laplace's equation in two dimensions is given by: —
∂²f/∂x² + ∂²f/∂y² = 0
5.
Laplace's equation in three dimensions is given by: —
∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z² = 0
6.
If f(x, y, z) is a harmonic function, then the divergence of its gradient is: —
∇ ⋅ ∇f
7.
If Vf(P) = (0, 0), then the directional derivative of f at P in any direction u is: —
0
8.
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)? —
2
9.
What is the Laplacian of the function f(x, y) = sin(x)cosh(y)? —
0
10.
What is the directional derivative of f(x, y) = 5 at the point (3, 4) in the direction of (1, 1)? —
0
11.
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? —
-3
12.
Consider the function f(x, y) = cos(x)sinh(y). What is its Laplacian? —
0
13.
What is the directional derivative of f(x, y) = √(x² + y²) at (3, 4) in the direction of the vector (0, 1)? —
4/5
14.
What is the directional derivative of f(x, y) = x² - y² at (2, 1) in the direction of the vector (3, 4)? —
8
15.
If Vf = (2x, 3y) at point (1, 2), what is the directional derivative of f in the direction u = (1/√2, 1/√2)? —
7/√2
16.
What is a harmonic function? —
A function that satisfies Laplace's equation.
17.
Which of the following is a consequence of the Maximum Modulus Principle for harmonic functions (related to the Maximum/Minimum Principle)? —
A non-constant harmonic function cannot attain its maximum or minimum value inside the domain.
18.
The directional derivative measures the rate of change of a function: —
Along a specific direction.
19.
If f is a harmonic function, then its Laplacian is: —
Zero
20.
If f is harmonic, then the integral of f over any closed curve C in its domain is: —
Zero
21.
If a function is harmonic, its second partial derivatives are: —
Continuous
22.
A function that is harmonic in a simply connected domain and has no singularities is called: —
Harmonic function
23.
Which of the following is NOT a common application of harmonic functions? —
Quantum mechanics wave functions
24.
Which of the following is an example of a harmonic function in 3D? —
f(x, y, z) = x² + y² - 2z²
25.
Which of the following functions is harmonic? —
f(x, y) = x² - y²
26.
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)? —
Mean Value Property
27.
The operator ∇² is known as the: —
Laplacian operator
28.
The set of all harmonic functions on a domain forms a: —
Vector space
29.
The function f(x, y) = e^x sin(y) is: —
Not harmonic
30.
If f(x, y) = ax + by + c, then f is: —
Harmonic only if a=b=0
31.
Which of the following is NOT a property of the directional derivative? —
It is always non-negative.
32.
Which of the following is a necessary condition for a function to be harmonic? —
Its Laplacian must be zero.
33.
The property that a harmonic function has no local maxima or minima in its domain (unless it is constant) is known as the: —
Maximum/Minimum Principle
34.
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: —
Both harmonic
35.
The directional derivative of a constant function is always: —
Zero
36.
If f is harmonic and u is a unit vector, then the directional derivative of f in the direction u is: —
The negative of the directional derivative of f in the direction -u.
37.
The directional derivative of f at P in the direction of u is zero if u is perpendicular to: —
The gradient of f at P.
38.
To find the maximum rate of increase of a function f at a point P, one should move in the direction of: —
The gradient of f at P.
39.
What is the definition of the directional derivative of a scalar function f(x, y, z) at a point P in the direction of a unit vector u? —
The rate of change of f at P in the direction of u, calculated as the dot product of the gradient of f at P and the unit vector u.
40.
The Mean Value Property for harmonic functions states that the average value of a harmonic function over a sphere is equal to: —
The value of the function at the center of the sphere.
41.
The function f(x, y) = x² - y² is harmonic in the entire xy-plane. —
True
42.
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. —
True
43.
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. —
True
44.
A function f is harmonic if and only if it is the real or imaginary part of an analytic function (in 2D). —
True
45.
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. —
True
46.
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: —
u ⋅ (Vf)
47.
For a function f(x, y, z), the directional derivative in the direction of vector v (not necessarily unit) is given by: —
(Vf ⋅ v) / ||v||
48.
Which physical phenomenon is often described by Laplace's equation? —
Diffusion of heat in steady state
49.
Consider f(x, y) = x³ - 3xy². Is f harmonic? —
Yes
50.
Consider the function f(x, y) = ln(x² + y²). Is this function harmonic? (Assume x² + y² ≠ 0) —
Yes, its Laplacian is 0.