/sci/ - Science & Math

I’m trying to figure out a section of the book “Environmental Soil Physics”, by Daniel Hillel (1998), namely the step from eq. (8.12)

[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\left(\Psi - z\right)\right][/eqn]

to eq. (8.12a)

[eqn]\frac{\partial \theta}{\partial t} = - \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] + \frac{\partial K}{\partial z}[/eqn]

I’ve only gotten this far:

[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\left(\Psi - z\right)\right][/eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \nabla z\right)\right][/eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \left(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} + \frac{\partial z}{\partial z}\right)\right)\right][/eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \left(0 + 0 + 1\right)\right)\right][/eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - 1\right)\right][/eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi - K\left(\Psi\right)\right][/eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] - \nabla \cdot K\left(\Psi\right)[/eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] - \left(\frac{\partial K\left(\Psi\right)}{\partial x} + \frac{\partial K \left(\Psi\right)}{\partial y} + \frac{\partial K \left(\Psi\right)}{\partial z}\right)[/eqn]
Can someone please point out to me what I’m missing?

I’m trying to figure out a section of the book “Environmental Soil Physics”, by Daniel Hillel (1998), namely the step from eq. (8.12)

[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\left(\Psi - z\right)\right][\eqn]

to eq. (8.12a)

[eqn]\frac{\partial \theta}{\partial t} = - \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] + \frac{\partial K}{\partial z}[\eqn]

I’ve only gotten this far:

[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\left(\Psi - z\right)\right][\eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \nabla z\right)\right][\eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \left(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} + \frac{\partial z}{\partial z}\right)\right)\right][\eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \left(0 + 0 + 1\right)\right)\right][\eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - 1\right)\right][\eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi - K\left(\Psi\right)\right][\eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] - \nabla \cdot K\left(\Psi\right)[\eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] - \left(\frac{\partial K\left(\Psi\right)}{\partial x} + \frac{\partial K \left(\Psi\right)}{\partial y} + \frac{\partial K \left(\Psi\right)}{\partial z}\right)[\eqn]
Can someone please point out to me what I’m missing?

I’m trying to figure out a section of the book “Environmental Soil Physics”, by Daniel Hillel (1998), namely the step from eq. (8.12)

[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\left(\Psi - z\right)\right][eqn]

to eq. (8.12a)

[eqn]\frac{\partial \theta}{\partial t} = - \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] + \frac{\partial K}{\partial z}[eqn]

I’ve only gotten this far:

[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\left(\Psi - z\right)\right][eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \nabla z\right)\right][eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \left(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} + \frac{\partial z}{\partial z}\right)\right)\right][eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \left(0 + 0 + 1\right)\right)\right][eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - 1\right)\right][eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi - K\left(\Psi\right)\right][eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] - \nabla \cdot K\left(\Psi\right)[eqn]
[eqn]\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] - \left(\frac{\partial K\left(\Psi\right)}{\partial x} + \frac{\partial K \left(\Psi\right)}{\partial y} + \frac{\partial K \left(\Psi\right)}{\partial z}\right)[eqn]
Can someone please point out to me what I’m missing?

I’m trying to figure out a section of the book “Environmental Soil Physics”, by Daniel Hillel (1998), namely the step from eq. (8.12)

[eqn]
\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\left(\Psi - z\right)\right]
[eqn]

to eq. (8.12a)

[eqn]
\frac{\partial \theta}{\partial t} = - \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] + \frac{\partial K}{\partial z}
[eqn]

I’ve only gotten this far:

[eqn]
\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\left(\Psi - z\right)\right]
[eqn]

[eqn]
\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \nabla z\right)\right]
[eqn]

[eqn]
\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \left(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} + \frac{\partial z}{\partial z}\right)\right)\right]
[eqn]

[eqn]
\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - \left(0 + 0 + 1\right)\right)\right]
[eqn]

[eqn]
\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\left(\nabla\Psi - 1\right)\right]
[eqn]

[eqn]
\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi - K\left(\Psi\right)\right]
[eqn]

[eqn]
\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] - \nabla \cdot K\left(\Psi\right)
[eqn]

[eqn]
\frac{\partial \theta}{\partial t} = \nabla \cdot \left[K\left(\Psi\right)\nabla\Psi\right] - \left(\frac{\partial K\left(\Psi\right)}{\partial x} + \frac{\partial K \left(\Psi\right)}{\partial y} + \frac{\partial K \left(\Psi\right)}{\partial z}\right)
[eqn]

Can someone please point out to me what I’m missing?