Divergence & Curl: Core Definitions, Notation & Geometric Insight

The correct formula is div(F) = ∂P/∂x + ∂Q/∂y + ∂R/∂z. Option B is the formula for the curl of F. Option C is a nonsensical combination of derivatives. Option D, ∂P/∂z + ∂Q/∂x + ∂R/∂y, does not correspond to any standard vector calculus operator, though it resembles the components of a gradient if the variables were permuted.

The curl of a vector field is a vector quantity that describes the infinitesimal rotation of the field at a point. It indicates the axis of rotation and the magnitude of the "circulation" or "twisting" strength of the field. A non-zero curl suggests the presence of rotation, like in a whirlpool or a swirling fluid. The other options describe properties related to the divergence (like expansion or source strength) or flux, not the rotational tendency measured by the curl.

First, we compute ∂P/∂x, which is the derivative of x² with respect to x, giving 2x. Next, we compute ∂Q/∂y, which is the derivative of 3xy with respect to y. Treating x as a constant, the derivative of 3xy with respect to y is 3x. Then, we compute ∂R/∂z, which is the derivative of z with respect to z, giving 1. Finally, we sum these results: 2x + 3x + 1 = 5x + 1. Therefore, the divergence is 5x + 1.

The curl of a vector field F = P i + Q j + R k is given by the determinant of a specific matrix, resulting in the vector: (∂R/∂y - ∂Q/∂z) i + (∂P/∂z - ∂R/∂x) j + (∂Q/∂x - ∂P/∂y) k. For F = y i + z j + x k, we have P = y, Q = z, and R = x. We calculate each component step by step. For the i-component: ∂R/∂y is the derivative of x with respect to y, which is 0. ∂Q/∂z is the derivative of z with respect to z, which is 1. So the i-component is 0 - 1 = -1. For the j-component: The formula is (∂P/∂z - ∂R/∂x). We have ∂P/∂z (the derivative of y with respect to z) which is 0. We have ∂R/∂x (the derivative of x with respect to x) which is 1. So the j-component is 0 - 1 = -1. For the k-component: ∂Q/∂x is the derivative of z with respect to x, which is 0. ∂P/∂y is the derivative of y with respect to y, which is 1. So the k-component is 0 - 1 = -1. Therefore, the curl is (-1) i + (-1) j + (-1) k.

Divergence measures the net rate of outward flux per unit volume from an infinitesimally small region around a point. A positive divergence means that more of the vector field is flowing out of the point than flowing into it. This indicates that the point is a "source" of the field, like a faucet emitting water. A negative divergence indicates a "sink," where the field flows inward. Divergence does not provide information about rotation.

The curl of a vector field representing fluid flow indicates the axis and intensity of rotation at a point. For the field F = (-y) i + (x) j, the curl is calculated as (∂Q/∂x - ∂P/∂y) k. Here, P = -y and Q = x. ∂Q/∂x = 1, ∂P/∂y = -1. So the k-component of the curl is 1 - (-1) = 2. This constant positive value (in the k-direction) indicates a consistent counterclockwise rotation (by the right-hand rule) throughout the fluid. The other options describe behaviors related to divergence (expansion/compression) or incorrectly state no rotation.

To find the divergence, we compute ∂P/∂x + ∂Q/∂y + ∂R/∂z, where P = x sin(y), Q = y ex, and R = xyz. First, ∂P/∂x: the derivative of x sin(y) with respect to x is sin(y) (treating sin(y) as constant). Next, ∂Q/∂y: the derivative of y ex with respect to y is ex (treating ex as a constant coefficient, since the derivative of y is 1). Then, ∂R/∂z: the derivative of xyz with respect to z is xy (treating x and y as constants). Summing these, we get sin(y) + ex + xy.

The curl is given by (∂R/∂y - ∂Q/∂z) i + (∂P/∂z - ∂R/∂x) j + (∂Q/∂x - ∂P/∂y) k. For F with P = z², Q = x², R = y², we compute each component. For the i-component: ∂R/∂y = derivative of y² with respect to y = 2y. ∂Q/∂z = derivative of x² with respect to z = 0 (since x² has no z). So i-component = 2y - 0 = 2y. The j-component of curl is ∂P/∂z - ∂R/∂x = 2z - 0 = 2z. For the k-component: ∂Q/∂x = derivative of x² with respect to x = 2x. ∂P/∂y = derivative of z² with respect to y = 0. So k-component = 2x - 0 = 2x. Therefore, the curl is (2y) i + (2z) j + (2x) k.

Divergence of a velocity field represents the net rate of outflow of fluid per unit volume from a point. A divergence of zero everywhere means that for any infinitesimal volume, the amount of fluid flowing in equals the amount flowing out. This condition defines an incompressible fluid, where the density remains constant. Additionally, it implies there are no sources (points where fluid is created) or sinks (points where fluid is destroyed) within the field. While zero divergence is related to incompressibility, it does not necessarily imply the fluid is irrotational (which would require zero curl). Therefore, both A and C are correct interpretations.

The curl of a vector field quantifies the rotational component or circulation density of the field at a point. In physics, for a fluid velocity field, a non-zero curl indicates the presence of local rotation, eddies, or vortex motion. The other options are more closely associated with other concepts: heat diffusion is related to the gradient and divergence, electric charge strength relates to divergence (Gauss's law), and pressure gradients relate to the gradient of a scalar field, not the curl of a vector field.

To find the divergence, we compute ∂P/∂x + ∂Q/∂y + ∂R/∂z. Here, P = 3xy, Q = y², R = z. ∂P/∂x = derivative of 3xy with respect to x = 3y. ∂Q/∂y = derivative of y² with respect to y = 2y. ∂R/∂z = derivative of z with respect to z = 1. Summing gives 3y + 2y + 1.

We compute the curl using (∂R/∂y - ∂Q/∂z) i + (∂P/∂z - ∂R/∂x) j + (∂Q/∂x - ∂P/∂y) k. Given P=3xy, Q=y², R=z. For the i-component: ∂R/∂y = derivative of z with respect to y = 0. ∂Q/∂z = derivative of y² with respect to z = 0. So i-component = 0 - 0 = 0. For the j-component: ∂P/∂z = derivative of 3xy with respect to z = 0. ∂R/∂x = derivative of z with respect to x = 0. So j-component = 0 - 0 = 0. For the k-component: ∂Q/∂x = derivative of y² with respect to x = 0. ∂P/∂y = derivative of 3xy with respect to y = 3x. So k-component = 0 - 3x = -3x. Therefore, the curl is (0) i + (0) j + (-3x) k.

The divergence of F is ∂P/∂x + ∂Q/∂y, where P = ax + by and Q = cx + dy. We compute ∂P/∂x = a and ∂Q/∂y = d. Therefore, divergence = a + d. For the divergence to be zero, we need a + d = 0. The other conditions relate to properties like being irrotational (zero curl) or harmonic, but not zero divergence.

For a 2D field with no z-component, the curl only has a k-component: curl(F)_k = ∂Q/∂x - ∂P/∂y. Here, P = y² and Q = x². ∂Q/∂x = 2x, ∂P/∂y = 2y. So curl = (0) i + (0) j + (2x - 2y) k.

This is a vector calculus identity: div(curl F) = 0 for any vector field F with continuous second partial derivatives. It means that the curl of a vector field is always divergence-free. Option B is nonsense because the divergence of F is a scalar, and you cannot take the curl of a scalar. Option C is false: the divergence of the gradient is the Laplacian, which is not necessarily zero. Option D is false: the curl of the gradient of any scalar function is always the zero vector.