Geometric arguments concerning Lagrange multipliers

[samspade]

I’m currently working through ‘mathematical methods in the physical sciences’ by Mary Boas and am having trouble with a detail of an argument in the book. I've attached the two relevant pages, the arguments proceeds roughly from equ 9.9 onwards. I think I understand the argument step by step all except one point regarding the phi function. If,

φ(x, y, z) = const

how can

grad φ

have any meaning as a vector given that from what I can tell it is equal to 0, I'm concerned that proceeding arguments concerning the proportionality of f and phi and the perpendicularity of the grad vectors with the tangent plane are meaningless as grad phi equals 0. If someone could educate me I'd be very appreciative.
Hope I'm clear enough, first time posting.
Thanks in advance,
Sam.

 

[HallsofIvy]

"how can grad φ have any meaning as a vector given that from what I can tell it is equal to 0" seems strange to me. grad φ is, by definition, a vector.at any point, (x, y, z). IF it is the 0 vector, then it is still a vector! But you are misunderstanding the what you link to. It does NOT say that φ(x, y z) is constant for all (x, y, z). It is saying that φ is constant along a given curve ("level curves" for φ). The "directional derivative" in the direction of that curve is 0. Since the directional derivative is the dot product of grad φ with a unit vector in that direction, it follows that grad φ is perpendicular to the "level curves".

 

[samspade]

Thanks so much or your time, I feel I’ve been so close to that for so long and now that I see how obvious it is it’s a little frustrating. I am aware of the vector 0 I was just imprecise in my wording. I’ll likely use this forum again