Divergence & Curl Identities Quiz

1) Compute the curl of F(x,y,z) = <3x, 4y, 5z>.

<0, 0, 0>

<5, 4, 3>

<0, 0, 12>

<4, -3, 0>

Using the curl formula: i (∂R/∂y − ∂Q/∂z) − j (∂R/∂x − ∂P/∂z) + k (∂Q/∂x − ∂P/∂y). Here R = 5z (no y), so ∂R/∂y = 0; Q = 4y (no z), so ∂Q/∂z = 0 → i component = 0 − 0 = 0. Similarly, ∂R/∂x = 0, ∂P/∂z = 0 → j component = −(0 − 0) = 0. ∂Q/∂x = 0, ∂P/∂y = 0 → k component = 0 − 0 = 0. So curl F = <0, 0, 0>.

Explanation

Using the curl formula: i (∂R/∂y − ∂Q/∂z) − j (∂R/∂x − ∂P/∂z) + k (∂Q/∂x − ∂P/∂y). Here R = 5z (no y), so ∂R/∂y = 0; Q = 4y (no z), so ∂Q/∂z = 0 → i component = 0 − 0 = 0. Similarly, ∂R/∂x = 0, ∂P/∂z = 0 → j component = −(0 − 0) = 0. ∂Q/∂x = 0, ∂P/∂y = 0 → k component = 0 − 0 = 0. So curl F = <0, 0, 0>.

2) Calculate the divergence of the one-dimensional vector field F = <5x, 0, 0>.

0

5x

5

<5, 0, 0>

The divergence is the sum of partial derivatives.

d/dx(5x) = 5.

d/dy(0) = 0.

d/dz(0) = 0.

Total divergence = 5 + 0 + 0 = 5.

Explanation

The divergence is the sum of partial derivatives.

d/dx(5x) = 5.

d/dy(0) = 0.

d/dz(0) = 0.

Total divergence = 5 + 0 + 0 = 5.

3) A fluid flows in the xy-plane rotating counter-clockwise around the origin. According to the right-hand rule, what is the direction of the curl vector?

Positive x-direction

Positive z-direction

Negative z-direction

Radially outward

Using the right-hand rule, if you curl the fingers of your right hand in the direction of the rotation (counter-clockwise), your thumb points up, which corresponds to the positive z-direction (the vector k).

Explanation

Using the right-hand rule, if you curl the fingers of your right hand in the direction of the rotation (counter-clockwise), your thumb points up, which corresponds to the positive z-direction (the vector k).

4) Consider the shear flow velocity field V = <y, 0, 0>. Calculate the curl of this field.

<0, 0, 0>

<0, 0, -1>

<0, 0, 1>

<1, 0, 0>

We compute the curl using the determinant method or the component formula. P = y, Q = 0, R = 0. The k-component is ∂Q/∂x - ∂P/∂y. Here, Q = 0 and P = y. So, ∂Q/∂x = ∂(0)/∂x = 0 and ∂P/∂y = ∂(y)/∂y = 1. Thus, the k-component is 0 - 1 = -1. The i and j components are zero. The result is < 0, 0, -1 >. This illustrates that a fluid moving in straight lines can still have a non-zero curl if the velocity varies perpendicular to the flow (shear), causing a paddle wheel to rotate clockwise.

Explanation

We compute the curl using the determinant method or the component formula. P = y, Q = 0, R = 0. The k-component is ∂Q/∂x - ∂P/∂y. Here, Q = 0 and P = y. So, ∂Q/∂x = ∂(0)/∂x = 0 and ∂P/∂y = ∂(y)/∂y = 1. Thus, the k-component is 0 - 1 = -1. The i and j components are zero. The result is < 0, 0, -1 >. This illustrates that a fluid moving in straight lines can still have a non-zero curl if the velocity varies perpendicular to the flow (shear), causing a paddle wheel to rotate clockwise.

5) In a tank of liquid rotating as a rigid body with angular velocity ω = 5 k (rad/s), what is the magnitude of the curl of the velocity field at any point inside the liquid?

0

5

10

Depends on distance

For rigid-body rotation with angular velocity vector ω = <0, 0, ω>, the velocity field is v = ω × r = <−ω y, ω x, 0>. The curl of this velocity field is always <0, 0, 2ω>, regardless of position. Here ω = 5, so |curl v| = |2ω| = 10 (in units of s⁻¹). The factor of 2 comes from the standard result for rigid rotation: curl v = 2ω.

Explanation

For rigid-body rotation with angular velocity vector ω = <0, 0, ω>, the velocity field is v = ω × r = <−ω y, ω x, 0>. The curl of this velocity field is always <0, 0, 2ω>, regardless of position. Here ω = 5, so |curl v| = |2ω| = 10 (in units of s⁻¹). The factor of 2 comes from the standard result for rigid rotation: curl v = 2ω.

6) When computing the divergence of the product of a scalar function f and a vector field F, denoted as div(fF), why is the result not simply f times div(F)?

Because divergence is linear

Because f varies spatially

Because result must be vector

Product rule does not apply

The derivative acts on both f and F, giving div(fF) = grad(f)·F + f div(F).

Explanation

The derivative acts on both f and F, giving div(fF) = grad(f)·F + f div(F).

7) True or False: For a two-dimensional vector field F = <P(x,y), Q(x,y)>, the scalar component of the curl is calculated as (partial of Q with respect to x) + (partial of P with respect to y).

True

False

The correct formula for the k-component (or scalar value) of the curl for a 2D field is (partial of Q with respect to x) minus (partial of P with respect to y). The statement incorrectly uses a plus sign instead of a minus sign. This is derived from the determinant of the matrix used to calculate the cross product.

Explanation

The correct formula for the k-component (or scalar value) of the curl for a 2D field is (partial of Q with respect to x) minus (partial of P with respect to y). The statement incorrectly uses a plus sign instead of a minus sign. This is derived from the determinant of the matrix used to calculate the cross product.

8) Why is the expression grad(div F) a valid vector operation, whereas div(grad F) is a valid scalar operation?

Because divergence outputs scalar and gradient acts on scalars

Because divergence outputs vector

Both result in vectors

Both are scalars

Operations must match input/output types. The divergence operator takes a vector and returns a scalar. The gradient operator takes a scalar and returns a vector. Therefore, grad(div F) takes Vector -> Scalar -> Vector. Conversely, div(grad f) takes Scalar -> Vector -> Scalar (this is the Laplacian).

Explanation

Operations must match input/output types. The divergence operator takes a vector and returns a scalar. The gradient operator takes a scalar and returns a vector. Therefore, grad(div F) takes Vector -> Scalar -> Vector. Conversely, div(grad f) takes Scalar -> Vector -> Scalar (this is the Laplacian).

9) If a smooth vector field F defined on all of 3D space has a non-zero curl, what can you definitively conclude about F?

It is conservative

It is not conservative

It has zero divergence

Line integrals are zero

A necessary condition for a vector field to be conservative (i.e., to have a potential function f such that F = grad f) is that its curl must be zero everywhere. If the curl is non-zero, the field possesses "circulation" and cannot be generated purely by a scalar potential.

Explanation

A necessary condition for a vector field to be conservative (i.e., to have a potential function f such that F = grad f) is that its curl must be zero everywhere. If the curl is non-zero, the field possesses "circulation" and cannot be generated purely by a scalar potential.

10) Consider a gas flowing through a pipe that runs along the x-axis. The velocity field is V = < u(x), 0, 0 >. If the gas is being compressed as it moves (density increases), what must be true about the derivative du/dx?

Du/dx must be positive.

Du/dx must be negative.

Du/dx must be zero.

Du/dx must equal the pressure.

Divergence measures the rate of outgoing flux per unit volume. For a gas flow V, a positive divergence implies expansion (density decrease), and negative divergence implies compression (density increase). The divergence of V = < u(x), 0, 0 > is simply du/dx. Therefore, for compression (negative divergence), du/dx must be negative. This physically means the gas in front is moving slower than the gas behind it, causing it to "pile up."

Explanation

Divergence measures the rate of outgoing flux per unit volume. For a gas flow V, a positive divergence implies expansion (density decrease), and negative divergence implies compression (density increase). The divergence of V = < u(x), 0, 0 > is simply du/dx. Therefore, for compression (negative divergence), du/dx must be negative. This physically means the gas in front is moving slower than the gas behind it, causing it to "pile up."

11) Consider the shear flow velocity field V = < y², 0, 0 >. At which locations in the xy-plane is the curl vector equal to zero?

Along y=x

Only at origin

Along x-axis

Nowhere

We calculate the curl of V = < P, Q, R > = < y², 0, 0 >. The k-component of the curl is dQ/dx - dP/dy. Here, dQ/dx = 0 and dP/dy = 2y. So the curl is < 0, 0, -2y >. For the curl to be the zero vector, each component must be zero. This requires -2y = 0, which implies y = 0. Thus, the curl is zero only along the x-axis.

Explanation

We calculate the curl of V = < P, Q, R > = < y², 0, 0 >. The k-component of the curl is dQ/dx - dP/dy. Here, dQ/dx = 0 and dP/dy = 2y. So the curl is < 0, 0, -2y >. For the curl to be the zero vector, each component must be zero. This requires -2y = 0, which implies y = 0. Thus, the curl is zero only along the x-axis.

12) Calculate the gradient of the divergence of the vector field F = < x³/3, y³/3, z³/3 >.

<2x,2y,2z>

<x²,y²,z²>

<6x,6y,6z>

0

First, find the divergence: d/dx(x³/3) + d/dy(y³/3) + d/dz(z³/3) = x² + y² + z². Now, take the gradient of this scalar: < d/dx(x²+...), d/dy(...+y²+...), d/dz(...) > = < 2x, 2y, 2z >.

Explanation

First, find the divergence: d/dx(x³/3) + d/dy(y³/3) + d/dz(z³/3) = x² + y² + z². Now, take the gradient of this scalar: < d/dx(x²+...), d/dy(...+y²+...), d/dz(...) > = < 2x, 2y, 2z >.

13) A vector field is given by F = < g(x), y, z >. Find the function g(x) such that the field has zero divergence everywhere, given the initial condition g(0) = 5.

G(x) = -2x + 5

G(x) = 2x + 5

G(x) = 5e^{-x}

G(x) = x² + 5

The divergence of F is d/dx(g(x)) + d/dy(y) + d/dz(z).

d/dy(y) = 1 and d/dz(z) = 1.

So, div F = g'(x) + 1 + 1 = g'(x) + 2.

For the divergence to be zero, we need g'(x) + 2 = 0, or g'(x) = -2.

Integrating with respect to x gives g(x) = -2x + C.

Using the condition g(0) = 5, we find C = 5.

Therefore, g(x) = -2x + 5.

Explanation

The divergence of F is d/dx(g(x)) + d/dy(y) + d/dz(z).

d/dy(y) = 1 and d/dz(z) = 1.

So, div F = g'(x) + 1 + 1 = g'(x) + 2.

For the divergence to be zero, we need g'(x) + 2 = 0, or g'(x) = -2.

Integrating with respect to x gives g(x) = -2x + C.

Using the condition g(0) = 5, we find C = 5.

Therefore, g(x) = -2x + 5.

14) Determine if the vector field F = < y, 0, 0 > is conservative by calculating its curl.

Yes, it is conservative because the curl is 0.

No, it is not conservative because the curl is not 0.

Yes, it is conservative because the divergence is 0.

Cannot be determined.

For a field to be conservative, its curl must be the zero vector. Let's calculate the k-component of the curl for F = < y, 0, 0 >: dQ/dx - dP/dy. Here P=y and Q=0.

dQ/dx = 0.

dP/dy = 1.

k-component = 0 - 1 = -1.

Since the curl is < 0, 0, -1 > (not zero), the field is not conservative.

Explanation

For a field to be conservative, its curl must be the zero vector. Let's calculate the k-component of the curl for F = < y, 0, 0 >: dQ/dx - dP/dy. Here P=y and Q=0.

dQ/dx = 0.

dP/dy = 1.

k-component = 0 - 1 = -1.

Since the curl is < 0, 0, -1 > (not zero), the field is not conservative.

15) Consider a radial vector field F = rⁿ r̂, where r is the distance from the origin, r̂ is the unit radial vector, and n is a constant. For what value of n is the divergence of this field equal to zero everywhere (except at the origin)?

N = -1

N = -2

N = -3

N = 0

We can write the field as F = rⁿ (rᵥec / r) = r^(n-1) rᵥec.

Using the divergence formula for a field g(r)rᵥec, which is ∇ · (g(r)rᵥec) = 3g(r) + r g'(r).

Here, g(r) = r^(n-1).

g'(r) = (n-1)r^(n-2).

Divergence = 3r^(n-1) + r(n-1)r^(n-2) = 3r^(n-1) + (n-1)r^(n-1) = (3 + n - 1)r^(n-1) = (n + 2)r^(n-1).

For the divergence to be zero, we must have n + 2 = 0, which implies n = -2.

Explanation

We can write the field as F = rⁿ (rᵥec / r) = r^(n-1) rᵥec.

Using the divergence formula for a field g(r)rᵥec, which is ∇ · (g(r)rᵥec) = 3g(r) + r g'(r).

Here, g(r) = r^(n-1).

g'(r) = (n-1)r^(n-2).

Divergence = 3r^(n-1) + r(n-1)r^(n-2) = 3r^(n-1) + (n-1)r^(n-1) = (3 + n - 1)r^(n-1) = (n + 2)r^(n-1).

For the divergence to be zero, we must have n + 2 = 0, which implies n = -2.

Divergence & Curl: Vector Identities, Product Rules & Physical Applications