Gradient, Divergence and Curl - Question Bank

1. If $F = \nabla \phi$ and $\phi$ is a harmonic function (i.e., $\nabla^2 \phi = 0$), then $F$ is both:
A) Irrotational and Solenoidal
B) Rotational and Divergent
C) Irrotational and Divergent
D) Rotational and Solenoidal
2. Which of the following is NOT a vector operator?
A) Gradient
B) Divergence
C) Curl
D) Scalar Field
3. What is the physical meaning of $\nabla \phi = 0$?
A) The scalar field is increasing at its maximum rate.
B) The scalar field is decreasing at its maximum rate.
C) The scalar field has a critical point (maximum, minimum, or saddle point).
D) The scalar field is constant everywhere.
4. If $\phi(x,y,z) = \frac{1}{r}$, what is $\nabla^2 \phi$ for $r \neq 0$?
A) $0$
B) $\frac{1}{r}$
C) $\frac{-1}{r^2}$
D) $\frac{-3}{r}$
5. If $\phi(x,y,z) = \frac{1}{r}$, where $r = \sqrt{x^2+y^2+z^2}$, what is $\nabla \phi$?
A) $\frac{-r}{r^3}$
B) $\frac{-r}{r^2}$
C) $\frac{-r}{r}$
D) $\frac{r}{r^3}$
6. The curl of the position vector $r = xi + yj + zk$ is:
A) $0$
B) $1$
C) $3$
D) $r$
7. The divergence of the position vector $r = xi + yj + zk$ is:
A) $0$
B) $1$
C) $3$
D) $r$
8. If $F(x, y, z) = yi + zj + xk$, what is $\nabla \times F$?
A) $i + j + k$
B) $y + z + x$
C) $j + k + i$
D) $i + j - k$
9. If $F(x, y, z) = xyzi$, what is $\nabla \cdot F$?
A) $xyz$
B) $yz$
C) $x$
D) $0$
10. Consider the vector field $F(x, y, z) = x^2i + y^2j + z^2k$. Calculate the magnitude of the gradient of $\phi(x,y,z) = x+y+z$ at $(1,1,1)$.
A) $\sqrt{3}$
B) $1$
C) $3$
D) $\sqrt{1^2+1^2+1^2}$
11. If $\phi(x, y, z) = x^3y^2z$, what is $\frac{\partial^2 \phi}{\partial x \partial y}$?
A) $3x^2y^2z$
B) $6x^2yz$
C) $3x^2y^2z$
D) $x^3y^2z$
12. The directional derivative of a scalar field $\phi$ in the direction of a unit vector $u$ is given by:
A) $\nabla \cdot u$
B) $\nabla \times u$
C) $u \cdot \nabla \phi$
D) $\nabla^2 \phi$
13. If $F(x, y, z) = \nabla \times A$, then $\nabla \cdot F$ is:
A) $\nabla^2 A$
B) 0
C) $\nabla \times A$
D) $\nabla \cdot A$
14. If $F(x, y, z) = \nabla \phi$, then $\nabla \cdot F$ is:
A) $\nabla \times \phi$
B) $\nabla^2 \phi$
C) $\nabla \phi$
D) 0
15. The divergence of the curl of any vector function is always:
A) Divergent
B) Irrotational
C) Zero
D) Non-zero
16. The curl of the gradient of any scalar function is always:
A) Divergent
B) Irrotational
C) Zero
D) Non-zero
17. If $F(x,y,z) = x^2i + y^2j + z^2k$, what is $\nabla \times F$?
A) $0$
B) $2xi + 2yj + 2zk$
C) $x^2i + y^2j + z^2k$
D) $(x^2+y^2+z^2)k$
18. If $F(x,y,z) = x^2yi + xy^2zj + xyz^2k$, what is $\nabla \cdot F$?
A) $2xy + y^2z + 2xyz$
B) $2xy + xy^2 + xyz^2$
C) $y^2z + x^2z + xy^2$
D) $2x y + x y^2 + x y z^2$
19. If $\phi(x,y,z) = e^{xyz}$, what is $\nabla \phi$?
A) $yz e^{xyz} i + xz e^{xyz} j + xy e^{xyz} k$
B) $e^{xyz} i + e^{xyz} j + e^{xyz} k$
C) $xyz e^{xyz}$
D) $e^{xyz}$
20. What is the physical interpretation of $\nabla \times F = 0$ for a vector field $F$?
A) The field is source-free.
B) The field is circulation-free.
C) The field is rotational.
D) The field is irrotational.
21. What is the physical interpretation of $\nabla \cdot F = 0$ for a vector field $F$?
A) The field is source-free.
B) The field is circulation-free.
C) The field is rotational.
D) The field is irrotational.
22. If $F(x, y, z) = x^2i + y^2j + z^2k$, calculate $\nabla \times F$.
A) $0i + 0j + 0k$
B) $2xi + 2yj + 2zk$
C) $x^2i + y^2j + z^2k$
D) $(x^2+y^2+z^2)k$
23. If $F(x, y, z) = xi + yj + zk$, calculate $\nabla \cdot F$ at the point $(1, 2, 3)$.
A) $1$
B) $2$
C) $3$
D) $6$
24. If $\phi(x, y, z) = 3x^2y - y^3z^2$, calculate $\frac{\partial \phi}{\partial x}$.
A) $6xy$
B) $3x^2$
C) $3x^2y$
D) $6xy - 3y^3z^2$
25. Stokes' theorem relates a surface integral of the curl of a vector field to a line integral of the vector field around the boundary of the surface. Which operator is involved in Stokes' theorem?
A) Gradient
B) Divergence
C) Curl
D) Laplacian
26. The divergence theorem relates a volume integral of the divergence of a vector field to a surface integral of the vector field over the boundary of the volume. Which operator is involved in the divergence theorem?
A) Gradient
B) Divergence
C) Curl
D) Laplacian
27. Which of the following identities is INCORRECT?
A) $\nabla \cdot (\nabla \times F) = 0$
B) $\nabla \times (\nabla \phi) = 0$
C) $\nabla \cdot (\nabla \phi) = \nabla \times \phi$
D) $\nabla \times (\nabla \times F) = \nabla(\nabla \cdot F) - \nabla^2 F$
28. If $F(x, y) = -yi + xj$, what is $\nabla \times F$?
A) $0$
B) $2k$
C) $-2k$
D) $i+j$
29. If $F(x, y) = xi + yj$, what is $\nabla \cdot F$?
A) $x+y$
B) $1+1$
C) $0$
D) $xi+yj$
30. If $\phi(x, y) = x^2 - y^2$, what is $\nabla \phi$?
A) $2xi - 2yj$
B) $2xi + 2yj$
C) $2x$
D) $-2y$
31. Consider the vector field $F(x, y, z) = yi - xj + zk$. Calculate $\nabla \times F$.
A) $0i + 0j + (-2k)$
B) $0i + 0j + 2k$
C) $i + j + k$
D) $yi - xj + zk$
32. Consider the vector field $F(x, y, z) = xy^2zi + x^2yzj + xyz^2k$. Calculate $\nabla \cdot F$.
A) $y^2z + x^2z + xy^2$
B) $y^2z + x^2z + xyz^2$
C) $2xyz + x^2z + xy^2$
D) $y^2z + x^2z + 2xyz$
33. If $F = \nabla \times A$, then $\nabla \cdot F$ is:
A) $\nabla^2 A$
B) 0
C) $\nabla \times A$
D) $\nabla \cdot A$
34. If $F = \nabla \phi$, then $\nabla \cdot F$ is:
A) $\nabla \times \phi$
B) $\nabla^2 \phi$
C) $\nabla \phi$
D) 0
35. The operator $\nabla^2$ is also known as:
A) Gradient operator
B) Divergence operator
C) Curl operator
D) Laplacian operator
36. The Laplacian of a scalar field $\phi$ is defined as:
A) $\nabla \phi$
B) $\nabla \cdot \nabla \phi$
C) $\nabla \times \nabla \phi$
D) $\nabla \cdot \phi$
37. If a vector field $F$ is solenoidal, then it can be expressed as the curl of a vector potential function, i.e., $F = \nabla \times A$. This implies:
A) $\nabla \phi = F$
B) $\nabla \cdot F = 0$
C) $\nabla \times F = 0$
D) $\nabla^2 F = 0$
38. If a vector field $F$ is irrotational, then it can be expressed as the gradient of a scalar potential function, i.e., $F = \nabla \phi$. This scalar potential function $\phi$ satisfies:
A) $\nabla \cdot F = 0$
B) $\nabla \times F = 0$
C) $\nabla^2 \phi = 0$
D) $\nabla \phi = 0$
39. If a scalar field $\phi$ is differentiable, then the curl of its gradient is always:
A) The gradient of $\phi$
B) The divergence of $\phi$
C) Zero
D) Non-zero
40. A vector field $F$ is called solenoidal if its divergence is:
A) Non-zero
B) Zero
C) Curl
D) Gradient
41. A vector field $F$ is called irrotational if its curl is:
A) Non-zero
B) Zero
C) Divergent
D) Gradient
42. If $F(x, y, z) = yi + xj + zk$, what is the curl of $F$?
A) $0i + 0j + 0k$
B) $i + j + k$
C) $yi + xj + zk$
D) $(x-y)i + (y-z)j + (z-x)k$
43. For a vector field $F = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k$, the curl is defined as:
A) $\nabla \cdot F$
B) $\nabla F$
C) $\nabla \times F$
D) $\nabla^2 F$
44. What does the curl of a vector field represent?
A) The rate of expansion or compression of the vector field.
B) The net outward flux of the vector field through a surface.
C) The tendency of the vector field to rotate or swirl around a point.
D) The magnitude of the vector field at a point.
45. If $F(x, y, z) = x^2i + y^2j + z^2k$, what is the divergence of $F$?
A) $2x + 2y + 2z$
B) $x^2 + y^2 + z^2$
C) $2xi + 2yj + 2zk$
D) $0$
46. For a vector field $F = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k$, the divergence is defined as:
A) $\nabla \times F$
B) $\nabla \cdot F$
C) $\nabla F$
D) $\nabla^2 F$
47. What does the divergence of a vector field represent?
A) The tendency of the vector field to rotate or swirl around a point.
B) The net outward flux of the vector field per unit volume from a small volume around a point.
C) The directional derivative of the vector field.
D) The line integral of the vector field around a closed curve.
48. If $\phi(x, y, z) = x^2y + y^2z$, what is the gradient of $\phi$?
A) $(2xy)i + (x^2 + 2yz)j + (y^2)k$
B) $(2xy)i + (y^2z)j + (x^2y)k$
C) $(2xy)i + (x^2 + 2yz)j + (y^2)k$
D) $(2xy)i + (x^2)j + (2yz)k$
49. For a scalar field $\phi(x, y, z)$, the gradient is defined as:
A) $\nabla \cdot \phi$
B) $\nabla \times \phi$
C) $\nabla \phi$
D) $\phi \nabla$
50. What does the gradient of a scalar field represent?
A) The rate of change of the scalar field along a specific direction.
B) The direction and magnitude of the greatest rate of increase of the scalar field.
C) The circulation of the scalar field around a point.
D) The flux of the scalar field through a surface.
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