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

Question

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.

Answer 1

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 $\operatorname{curl} A = \nabla \times A$).

Answer 2

If F is any scalar function ,then Grad F will be the vector one as gradient operator is a vectorial operator. In the said equation,you have a vector quantity in the left hand side.It means your right hand side must represent a vector quantity.A unit vector is there in the right hand side,so all other parts should be scalar.That's why magnitude of Grad F is taken there.It just represent a number(scalar).