Divergence and Curl
Divergence measures local net outward flux per unit volume. Curl measures local circulation and rotation. Compute both from partial derivatives and use curl to test conservative fields.
What a learner can do afterwards
- Compute the divergence and curl of a given field
- Describe what each means physically at a point
- Justify zero curl for smooth gradient fields and apply the converse with its domain conditions
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For a differentiable field F = (P, Q, R), P_x means the partial derivative of P with respect to x. Other subscripts work the same way. Divergence is the scalar div F = P_x + Q_y + R_z. Differentiate each component in its matching coordinate, holding the other coordinates fixed, then add. A term independent of that coordinate contributes zero.
Divergence is the limit of net outward flux through a shrinking closed surface divided by its enclosed volume. Positive divergence means local net outflow; negative means local net inflow. Zero means balance in this limit, not that the field is zero. For a velocity field, divergence describes local expansion or compression. It does not mean fluid is being created or destroyed at a point.
Curl is the vector curl F = (R_y - Q_z, P_z - R_x, Q_x - P_y). Find the six partial derivatives, then subtract in this order. For a plane field (P, Q, 0) independent of z, only the third component remains. It is Q_x - P_y, the expression you used in Green's circulation theorem.
For a differentiable velocity field, half the curl is the angular-velocity vector of the local rotational part. Curl's magnitude is twice that angular speed. Its direction follows the right-hand rule: point your right thumb along curl and your curled fingers show the rotation. Positive z means counterclockwise when viewed from positive z toward the origin. Local rotation is distinct from a particle's path around a distant point.
Divergence does not determine curl. For nonzero constants a and b, (ax, ay, az) has divergence 3a and zero curl; (-by, bx, 0) has zero divergence and curl (0, 0, 2b). A constant field has both zero. Differentiate sums term by term when combining these patterns. Arrow directions alone do not settle the calculation: parallel shear (y, 0, 0) has curl (0, 0, -1).
Differentiate before substituting a point. For F = (xy, y squared, z squared), the matching derivatives are y, 2y and 2z, so div F = 3y + 2z. At (0, 1, 2), this is 3 + 4 = 7. Hold the other coordinates fixed when differentiating products. If a coefficient is unknown, compute divergence first, set it equal to the required value and solve for the coefficient.
A conservative field is a gradient F = (f_x, f_y, f_z). If f has continuous second partial derivatives, differentiating in either order gives equal mixed partials. A second subscript means differentiate again. Substitution into the curl formula makes every component a difference of equal mixed partials. Each difference is zero, proving that this gradient field has zero curl.
For a field with continuous first partial derivatives, zero curl guarantees a potential on a connected open domain where every closed loop can shrink to a point within the domain. Without that domain condition, zero curl alone may not be sufficient. Check the domain as well as the derivatives: an excluded set can prevent loops from shrinking inside the domain.
Outward flux adds the normal component across a closed surface. Circulation adds the tangential component around a closed curve. For smooth fields and suitable boundaries, the divergence theorem gives outward flux as the volume integral of divergence. Stokes' theorem gives circulation as the surface integral of curl dotted with the unit normal over a spanning surface. The surface normal and boundary direction must have matching orientations.
Divergence measures local net outward flux per unit volume. Curl measures local circulation; for velocity fields, half the curl gives local angular velocity.
Lesson, questions, worked solutions and flashcards written for this dataset.
Readings, additional questions, worked solutions and flashcards written for this dataset.
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Keeping track of the derivatives
Write F = (P, Q, R). In P_x, the subscript tells you to differentiate P with respect to x while holding the other coordinates fixed.
To find divergence, use the matching coordinates: P_x + Q_y + R_z. For F = (2x + y, y - z, -z), the three derivatives are 2, 1 and -1. They add to 2. The extra terms y and -z contribute zero in those matching derivatives.
Curl uses six derivatives in a different arrangement: (R_y - Q_z, P_z - R_x, Q_x - P_y). Keep each subtraction in its stated order. Divergence returns a scalar, while curl returns a vector.
For a value at a point, calculate the derivative expression before inserting the coordinates. Substituting first can turn a variable expression into a constant and lose the derivative you need.
Motion without expansion
A nonzero velocity does not require nonzero divergence. For a constant velocity field, every partial derivative is zero. Fluid can move through a small region while the amounts entering and leaving balance.
Divergence records a local limit: net outward flux through a closed surface, divided by enclosed volume, as the region shrinks. Positive values indicate expansion for a velocity field; negative values indicate compression.
Rotation is another question. The field (-4y, 4x, 0) has zero divergence and curl (0, 0, 8). Half its curl gives the angular-velocity vector of its rotational part. Its angular speed is 4, counterclockwise when viewed from positive z toward the origin.
Visible loops are not required for curl. In (5y, 0, 0), arrows are parallel, but their changing lengths give curl (0, 0, -5). Derivatives reveal the local rotation that arrow directions alone can miss.
What the zero-curl test needs
Suppose F is the gradient of a potential f with continuous second partial derivatives. The x component of curl F subtracts the z derivative of f_y from the y derivative of f_z. These are equal mixed partial derivatives, so their difference is zero. The other two components cancel in the same way.
This proves a necessary condition: such a gradient has zero curl. To use zero curl in the reverse direction, check the field and its domain.
A field with continuous first partial derivatives and zero curl has a potential on a connected open domain where every closed loop can shrink to a point without leaving the domain. That loop condition matters because a local derivative test does not by itself settle a global question. If the condition is absent, zero curl alone may not decide whether a global potential exists.
Two boundary measurements
In Green's circulation theorem, the expression Q_x - P_y is integrated over a plane region. Extend (P, Q) to (P, Q, 0), independent of z. That expression becomes the z component of curl.
Circulation measures the field's tangential contribution around a closed curve. Outward flux measures its outward normal contribution across a closed surface. They use different components of the field and different boundaries.
For suitable smooth fields and boundaries, the divergence theorem equates outward flux with a volume integral of divergence. Stokes' theorem equates circulation with a surface integral of curl dotted with the oriented unit normal. The surface spans the boundary curve, and their orientations must agree.
These theorems turn sums of local derivatives into boundary totals. They do not say that one derivative value at one point replaces an entire integral.
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