I quote from the course material "The most important use for the tangent plane is to give an approximation that is the
basic formula in the study of functions of several variables — almost everything follows in
one way or another from it."

And yet, the fundamental equation they derive is incorrect.

A(x − x0) + B(y − y0) + C(w − w0) = 0 somehow becomes w − w0 = a(x − x0) + b(y − y0)?

Would someone mind explaining this magic? I've coped with the errors in every single PDF so far, but this one takes the cake, since the entire course is based upon that mystical rearrangement.

Question

I quote from the course material "The most important use for the tangent plane is to give an approximation that is the
basic formula in the study of functions of several variables — almost everything follows in
one way or another from it."

And yet, the fundamental equation they derive is incorrect.

A(x − x0) + B(y − y0) + C(w − w0) = 0 somehow becomes w − w0 = a(x − x0) + b(y − y0)?

Would someone mind explaining this magic? I've coped with the errors in every single PDF so far, but this one takes the cake, since the entire course is based upon that mystical rearrangement.

unknownunknown · Answer

http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-a-functions-of-two-variables-tangent-approximation-and-optimization/session-27-approximation-formula/MIT18_02SC_MNotes_ta2_3.pdf

irishboy123 · Answer

general plane is: $A(x − x_0) + B(y − y_0) + C(w − w_0) = 0$
assume: $C \ne 0$ ; why bother?
but if so, then: $\frac{A}{C}(x − x_0) + \frac{B}{C}(y − y_0) + (w − w_0) = 0$
$\frac{A}{C}(x − x_0) + \frac{B}{C}(y − y_0) = -(w − w_0)$
$w − w_0 = -A/C(x − x_0) + -B/C(y − y0)$

so the signs are incorrect, true...

but at this point, not sure i'd care.  the normal vector can point in either direction.  eg does this cause problems with the examples they pose?!  haven't read beyond the stuff mentioned herein.

not ideal.

unknownunknown · Answer

I'm just finding it difficult with the constant errors in the PDF's to know what's correct and what isn't. Hasn't been the first time in this course I've learnt something later to realize it's an error. For instance the minus sign can change the entire visualization direction. Yes true the direction doesn't matter, but still.. the lack of proof-checking is surprising for such an institution as MIT.

Now for this equation, I am confused what the normal components look like geometrically when they're divided by the z component. Eg: w/z, y/z. I need further explanation as to how the normal being divided by its z component is equivalent to the slope of the tangent at the point.

phi · Answer

This is a bit discursive, but perhaps it's helpful.

unknownunknown · Answer

@phi that's nice and helpful, thanks