# What does the notation $|\text{grad} \ F|$ mean?

I am currently studying the textbook Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edition, by Max Born and Emil Wolf. Page 5, chapter 1.1.3 Boundary conditions at a surface of discontinuity, says the following as a note at the bottom of the page:

$$^*$$ For later purposes we note a representation of the surface charge density and the surface current density in terms of the Dirac delta function (see Appendix IV). If the equation of the surface of discontinuity is $$F(x, y, z) = 0$$, then $$\rho = \hat{\rho} \mid \text{grad} \ F \mid \delta(F) \tag{17a}$$ $$\mathbf{j} = \hat{\mathbf{j}} \mid \text{grad} \ F \mid \delta(F) \tag{18a}$$ These relations can immediately be verified by substituting from (17a) and (18a) into (17) and (18) and using the relation $$dF = |\text{grad} \ F| dh$$ and the shifting property of the delta function.

Can someone please explain to me what this notation with $$|\text{grad} \ F|$$ means? I have never encountered it before.

Thank you.

• $grad F$ is a vector (the gradient of F) and |grad F| is the magnitude of that vector. Commented Jul 14, 2020 at 10:12
• @planetmaker Oh, so 17a just means $\hat{\rho}$ times the magnitude of $\text{grad} \ F$ times $\delta(F)$? Commented Jul 14, 2020 at 10:15
• To add to @planetmaker's comment, an alternative notation would be $|\vec{\nabla} F|$ Commented Jul 14, 2020 at 10:17
• @BySymmetry Ok, thanks. The form that (17a) and (18a) are written in almost makes it look like some new notation, rather than the magnitude. Commented Jul 14, 2020 at 10:18
• It looks totally normal and is very common notation.Only nit-pick is that I'd expect the magnitude bars closer to it's vector and not evenly spaced between the constituents of the equation. $\rho = \hat{\rho} |grad{F}| \delta(F)$ Commented Jul 14, 2020 at 10:42

It's the magnitude of the gradient of $$F$$.
$$\operatorname{grad} F$$ or $$\nabla F$$ denotes the gradient of the field F and thus $$|\operatorname{grad} F|$$ is the magnitude of the gradient of F.
It's common practise, but just a matter of taste, to use $$\operatorname{grad} F$$ over $$\nabla F$$ in order to guide the eye easier as to what is being done or used (similar to $$\operatorname{curl} A = \nabla \times A$$).
• B & W Is a wonderful book that was written before the notation of using $\nabla$ for 'grad' was adopted. Commented Jul 14, 2020 at 12:41
• Let me know when you get to $rot$. Commented Jul 14, 2020 at 15:03