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