Calculus
Divergence and curl
- What is divergence?
Divergence measures how much a vector field spreads out or converges at a point. It’s a scalar quantity that describes the “source” or “sink” strength of the field.
\[\nabla \cdot \mathbf{F} = \frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z}\]Physical interpretation:
Positive divergence: Field flows outward (source)
Negative divergence: Field flows inward (sink)
Zero divergence: No net outflow (incompressible flow)
Visualization of positive, zero, and negative divergence:
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Source code,png,hires.png,pdf)
The left panel shows positive divergence (source) for the vector field \(\mathbf{F} = (x, y)\), which has \(\nabla \cdot \mathbf{F} = +2\). Field lines radiate outward from the origin. The middle panel shows zero divergence for the rotational field \(\mathbf{F} = (-y, x)\), which has \(\nabla \cdot \mathbf{F} = 0\)—this is an incompressible flow where fluid circulates without expansion or compression. The right panel shows negative divergence (sink) for \(\mathbf{F} = (-x, -y)\), which has \(\nabla \cdot \mathbf{F} = -2\). Field lines converge toward the origin.
How to calculate divergence:
For the positive divergence case \(\mathbf{F} = (x, y)\):
\[\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} = \frac{\partial (x)}{\partial x} + \frac{\partial (y)}{\partial y} = 1 + 1 = 2\]For the zero divergence case \(\mathbf{F} = (-y, x)\):
\[\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} = \frac{\partial (-y)}{\partial x} + \frac{\partial (x)}{\partial y} = 0 + 0 = 0\]For the negative divergence case \(\mathbf{F} = (-x, -y)\):
\[\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} = \frac{\partial (-x)}{\partial x} + \frac{\partial (-y)}{\partial y} = -1 + (-1) = -2\]- What is curl?
Curl measures the rotation or circulation of a vector field. It’s a vector quantity that describes how much and in what direction the field “swirls” around a point.
\[\begin{split}\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ f_x & f_y & f_z \end{vmatrix}\end{split}\]\[= \mathbf{\hat{i}}\left(\frac{\partial f_z}{\partial y} - \frac{\partial f_y}{\partial z}\right) - \mathbf{\hat{j}}\left(\frac{\partial f_z}{\partial x} - \frac{\partial f_x}{\partial z}\right) + \mathbf{\hat{k}}\left(\frac{\partial f_y}{\partial x} - \frac{\partial f_x}{\partial y}\right)\]Visualization of positive, zero, and negative curl:
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Source code,png,hires.png,pdf)
The left panel shows positive curl (counterclockwise rotation) for the vector field \(\mathbf{F} = (-y, x)\), which has \(\nabla \times \mathbf{F} = +2\). The middle panel shows zero curl for the gradient field \(\mathbf{F} = (2x, 2y) = \nabla(x^2 + y^2)\), which has \(\nabla \times \mathbf{F} = 0\)—this is a conservative field that can be integrated to give a scalar potential. The right panel shows negative curl (clockwise rotation) for \(\mathbf{F} = (y, -x)\), which has \(\nabla \times \mathbf{F} = -2\).
How to calculate curl in 2D:
In 2D, we only need the z-component of curl (the component pointing out of the page):
\[(\nabla \times \mathbf{F})_z = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\]For the positive curl case \(\mathbf{F} = (-y, x)\):
\[(\nabla \times \mathbf{F})_z = \frac{\partial (x)}{\partial x} - \frac{\partial (-y)}{\partial y} = 1 - (-1) = 1 + 1 = 2\]For the zero curl case \(\mathbf{F} = (2x, 2y)\):
\[(\nabla \times \mathbf{F})_z = \frac{\partial (2y)}{\partial x} - \frac{\partial (2x)}{\partial y} = 0 - 0 = 0\]For the negative curl case \(\mathbf{F} = (y, -x)\):
\[(\nabla \times \mathbf{F})_z = \frac{\partial (-x)}{\partial x} - \frac{\partial (y)}{\partial y} = -1 - 1 = -2\]- What does “conservative field” mean?
A conservative field is a vector field that can be written as the gradient of a scalar function (potential). If \(\mathbf{F} = \nabla \phi\) for some scalar \(\phi\), then the field is conservative.
Key property: The curl of any gradient is zero:
\[\nabla \times (\nabla \phi) = \mathbf{0}\]This works both ways:
Forward: If you have a scalar function \(\phi(x,y)\), its gradient \(\nabla \phi = (\partial \phi/\partial x, \partial \phi/\partial y)\) gives you a vector field with zero curl
Backward (integration): If you have a vector field with zero curl, you can integrate it to find the original scalar function \(\phi\)
Example from the middle panel:
Start with scalar: \(\phi(x,y) = x^2 + y^2\)
Take gradient: \(\nabla \phi = (2x, 2y)\) — this is the vector field (arrows pointing outward)
Check curl: \(\nabla \times (2x, 2y) = \partial(2y)/\partial x - \partial(2x)/\partial y = 0 - 0 = 0\) ✓
Why “conservative”? The term comes from physics—specifically from the conservation of mechanical energy. When a force field is conservative, the total mechanical energy (kinetic + potential) remains constant as you move through the field.
The physics origin:
When you move an object through a force field \(\mathbf{F}\) from point A to point B, the work done is:
\[W = \int_A^B \mathbf{F} \cdot d\mathbf{r}\]For a conservative field where \(\mathbf{F} = -\nabla U\) (force is negative gradient of potential energy \(U\)):
\[W = \int_A^B (-\nabla U) \cdot d\mathbf{r} = -(U_B - U_A) = U_A - U_B\]The work depends only on the start and end points, not on the path taken. This means:
Closed path work is zero: If you return to your starting point (A = B), then \(W = U_A - U_A = 0\). No net energy is gained or lost going around a loop.
Energy is conserved: The work done by the force equals the decrease in potential energy, so \(E_{\text{kinetic}} + U = \text{constant}\).
Path independence: You can take any route from A to B and do the same amount of work.
Examples of conservative forces:
Gravity: \(\mathbf{F} = -mg\mathbf{\hat{z}}\), potential \(U = mgz\). Lifting an object 10 meters requires the same energy whether you go straight up or spiral around.
Electrostatic: \(\mathbf{F} = k\frac{q_1 q_2}{r^2}\mathbf{\hat{r}}\), potential \(U = k\frac{q_1 q_2}{r}\). Moving a charge in an electric field conserves energy.
Spring force: \(\mathbf{F} = -kx\mathbf{\hat{x}}\), potential \(U = \frac{1}{2}kx^2\). Compressing a spring stores energy that can be fully recovered.
Non-conservative forces:
Friction: Sliding an object around a loop dissipates energy as heat. The work depends on path length, not just endpoints. Cannot be written as \(\mathbf{F} = -\nabla U\).
Magnetic force: \(\mathbf{F} = q\mathbf{v} \times \mathbf{B}\). Always perpendicular to motion, does zero work, but still has curl (circulation around magnetic field lines).
- Why curl = 0 for conservative fields?
If \(\mathbf{F} = -\nabla U\), then by the vector calculus identity \(\nabla \times (\nabla U) = \mathbf{0}\), the curl must be zero. Physically, curl represents circulation or rotation in the field. If there’s circulation, you can extract energy by going around a loop (like a water wheel in a vortex), violating energy conservation.
Connection to DPC: When you measure center of mass shifts \((g_x, g_y)\) in DPC, you’re measuring the gradient of the phase \(\nabla \phi\). If curl is zero, you can integrate \(g_x\) and \(g_y\) to recover the original phase \(\phi(x,y)\). If curl is non-zero, the field is not conservative and integration gives inconsistent results (you can’t find a single-valued \(\phi\)).
- Why is curl important in DPC?
In Differential Phase Contrast (DPC) microscopy, we measure the center of mass shift vector field \(\mathbf{g}(r_x, r_y) = (\text{com}_x, \text{com}_y)\). This vector field should ideally be the gradient of a scalar phase:
\[\mathbf{g} = \nabla \phi = \left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}\right)\]In the ideal case, the curl is zero:
\[\nabla \times \mathbf{g} = \frac{\partial g_y}{\partial x} - \frac{\partial g_x}{\partial y} \approx 0\]This is because the curl of a gradient is always zero:
\[\nabla \times (\nabla f) = \mathbf{0}\]This is a fundamental vector calculus identity: the curl of any conservative vector field (one that can be written as the gradient of a function) is zero.
- What does it mean when curl is not zero in DPC?
When \(\nabla \times \mathbf{g} \neq 0\), the measured center of mass field is not conservative—it cannot be perfectly integrated to give a single-valued phase. This happens due to:
Measurement noise: Random errors in center of mass detection
Sample rotation: Physical misalignment between scan and detector coordinates
Non-ideal imaging conditions: Aberrations, multiple scattering
Software like Ophus computes the curl and finds the optimal rotation angle where \(\nabla \times \mathbf{g}\) becomes minimized. This aligns the coordinate systems and removes systematic errors before phase integration.
- What is the phase from electrostatic potential?
The phase shift acquired by an electron passing through an electrostatic potential \(V(x,y,z)\) is:
\[\phi(x,y) = \sigma \int V(x,y,z)\,dz\]where \(\sigma\) is the interaction constant. This is the real-space reconstructed phase image that relates to the sample’s potential.
“Phase contrast” means large \(\phi\) corresponds to regions with stronger potential (higher atomic number \(Z\) or greater thickness).
- Properties of divergence and curl summary
Key identities:
\(\nabla \times (\nabla f) = \mathbf{0}\) — curl of a gradient is zero
\(\nabla \cdot (\nabla \times \mathbf{F}) = 0\) — divergence of a curl is zero
If \(\nabla \times \mathbf{F} = \mathbf{0}\) everywhere, then \(\mathbf{F} = \nabla \phi\) for some scalar \(\phi\) (conservative field)
These identities are fundamental to understanding why DPC works: we measure gradients (center of mass shifts) and integrate them back to get phase, but this only works perfectly when curl is zero.
