Divergence & Curl: Fluid Flow, Sources, Sinks & Rotational Motion

The divergence is ∂(-x)/∂x + ∂(-y)/∂y + ∂(-z)/∂z = -1 + (-1) + (-1) = -3. A negative divergence indicates that the vector field is converging toward the point, meaning the point acts as a "sink" where fluid volume is being compressed or removed.

The direction of the curl vector follows the right-hand rule: if you curl the fingers of your right hand in the direction of the field's rotation, your thumb points in the direction of the curl vector. Since the curl points in the positive z-direction, the rotation must be in the xy-plane with a counterclockwise orientation (when viewed from the positive z-axis looking down). The magnitude of the curl vector is equal to twice the angular velocity of the rotation. Given that the curl magnitude is 4, we have angular velocity = curl magnitude / 2 = 4 / 2 = 2. Therefore, the field exhibits a counterclockwise rotation in the xy-plane with an angular velocity of 2

We compute the curl components. i-component: ∂(y)/∂y - ∂(-z)/∂z = 1 - (-1) = 2. j-component: ∂(0)/∂z - ∂(y)/∂x = 0 - 0 = 0. k-component: ∂(-z)/∂x - ∂(0)/∂y = 0 - 0 = 0. The curl is < 2, 0, 0 >, indicating rotation about the x-axis with magnitude 2.

A vector field with zero curl everywhere is called irrotational. Stokes' theorem relates the curl of a vector field to the circulation around a closed loop: ∮_C F·dr = ∬ₛ (∇×F)·n dS, where C is a closed curve, S is a surface bounded by C, and n is the unit normal. If ∇×F = 0 everywhere, then the right side of Stokes' theorem is zero for any surface S, which means the circulation ∮_C F·dr must be zero for any closed curve C. This is the precise meaning of statement B. Statement A is false because a field can have zero curl but still vary in magnitude and direction (e.g., F = <x, y, z> has zero curl but is not constant). Statement C is false because the curl's direction is not necessarily perpendicular to the field flow. Statement D is false because a field with zero curl is only guaranteed to be a gradient field if the domain is simply connected; on a domain with holes, a field can have zero curl but still not be conservative.

This is the product rule for divergence. The divergence of a scalar times a vector is the gradient of the scalar dotted with the vector, plus the scalar times the divergence of the vector. It is analogous to the product rule in single-variable calculus: (uv)' = u'v + uv'.

Divergence is calculated as the sum of partial derivatives, which is algebraically equivalent to the dot product of the vector operator ∇ and the vector field F. The notation ∇ × F represents the curl (cross product), and ∇ F usually represents the gradient of a scalar, not a vector field.

Curl measures the rotation of a vector field. It is formally defined as the cross product of the del operator (∇) with the vector field F. This cross product gives the vector whose components are exactly the differences of partial derivatives shown: the i-component is ∂R/∂y − ∂Q/∂z, the j-component is ∂P/∂z − ∂R/∂x, and the k-component is ∂Q/∂x − ∂P/∂y. This is the standard formula for curl F.

∂/∂x (xy) = y, ∂/∂y (yz) = z, ∂/∂z (zx) = x. Adding these gives y + z + x.

Using the formula for curl: i-component = ∂(y)/∂y - ∂(x)/∂z = 1 - 0 = 1. j-component = ∂(z)/∂z - ∂(y)/∂x = 1 - 0 = 1. k-component = ∂(x)/∂x - ∂(z)/∂y = 1 - 0 = 1. The result is the constant vector < 1, 1, 1 >.

The divergence is ∂/∂x (x²) + ∂/∂y (y²) + ∂/∂z (z²). The partial derivative of x² with respect to x is 2x, the partial derivative of y² with respect to y is 2y, and the partial derivative of z² with respect to z is 2z. Adding these gives 2x + 2y + 2z.

The curl of a vector field F = <P, Q, R> is given by ∇×F = <∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y>. For the field F(x, y, z) = <eˣ, eʸ, eᶻ>, we have P = eˣ, Q = eʸ, and R = eᶻ. We compute each partial derivative needed. For the first component: ∂R/∂y = ∂/∂y(eᶻ) = 0 because eᶻ does not depend on y, and ∂Q/∂z = ∂/∂z(eʸ) = 0 because eʸ does not depend on z. So the first component is 0 - 0 = 0. For the second component: ∂P/∂z = ∂/∂z(eˣ) = 0 because eˣ does not depend on z, and ∂R/∂x = ∂/∂x(eᶻ) = 0 because eᶻ does not depend on x. So the second component is 0 - 0 = 0. For the third component: ∂Q/∂x = ∂/∂x(eʸ) = 0 because eʸ does not depend on x, and ∂P/∂y = ∂/∂y(eˣ) = 0 because eˣ does not depend on y. So the third component is 0 - 0 = 0. Therefore, the curl is ∇×F = <0, 0, 0>. This zero result indicates that the vector field is conservative and represents an irrotational flow, meaning there is no local spinning motion at any point in the field.

The curl is computed using the formula ∇×F = <∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y>. For F(x, y, z) = <-y, x, 0>, we identify P = -y, Q = x, and R = 0. We calculate each partial derivative. For the first component: ∂R/∂y = ∂/∂y(0) = 0 and ∂Q/∂z = ∂/∂z(x) = 0 because x is constant with respect to z. So the first component is 0 - 0 = 0. For the second component: ∂P/∂z = ∂/∂z(-y) = 0 because -y is constant with respect to z, and ∂R/∂x = ∂/∂x(0) = 0. So the second component is 0 - 0 = 0. For the third component: the formula is ∂Q/∂x - ∂P/∂y. Here, ∂Q/∂x = ∂/∂x(x) = 1 and ∂P/∂y = ∂/∂y(-y) = -1. So the k-component is 1 - (-1) = 2. Therefore, the curl is ∇×F = <0, 0, 2>. The curl points entirely in the z-direction, which is consistent with a rotation about the z-axis. The magnitude of the curl, 2, represents twice the angular velocity of the rotation.

A field is solenoidal if its divergence is zero. We calculate the divergence of F: ∂(kx)/∂x + ∂(-4y)/∂y + ∂(2z)/∂z = k - 4 + 2 = k - 2. For the divergence to be zero, we must have k - 2 = 0, which implies k = 2.

A conservative vector field is the gradient of a scalar potential. A fundamental identity in vector calculus is that the curl of a gradient is always the zero vector (∇ × (∇f) = 0). Therefore, conservative fields are irrotational.

This question asks about a fundamental identity in vector calculus: the divergence of a curl is always zero. For any smooth vector field F = <P, Q, R>, we have ∇·(∇×F) = 0. To understand why, let's consider the curl first: ∇×F = <∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y>. Taking the divergence of this result means computing ∂/∂x of the first component plus ∂/∂y of the second component plus ∂/∂z of the third component. This gives us ∂/∂x(∂R/∂y - ∂Q/∂z) + ∂/∂y(∂P/∂z - ∂R/∂x) + ∂/∂z(∂Q/∂x - ∂P/∂y). Expanding this, we get ∂²R/∂x∂y - ∂²Q/∂x∂z + ∂²P/∂y∂z - ∂²R/∂y∂x + ∂²Q/∂z∂x - ∂²P/∂z∂y. By Clairaut's theorem on equality of mixed partials (for smooth functions), we have ∂²R/∂x∂y = ∂²R/∂y∂x, ∂²Q/∂x∂z = ∂²Q/∂z∂x, and ∂²P/∂y∂z = ∂²P/∂z∂y. Substituting these equalities, we see that each term cancels with another: ∂²R/∂x∂y cancels with -∂²R/∂y∂x, -∂²Q/∂x∂z cancels with ∂²Q/∂z∂x, and ∂²P/∂y∂z cancels with -∂²P/∂z∂y. The final result is identically zero.