Difference between Tangent Plane and Linearization

[newyorkcity]

What is the difference?

According to my text...

Tangent Plane:

z-z₀=f_x(x0,y0)(x-x0) + f_y(x0,y0)(y-y0)


Linearization:

L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)

 

[newyorkcity]

What is the difference?

According to my text...

Tangent Plane:

z-z0=fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)


Linearization:

L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)

 

[anonymity]

Well, think back to 2-dimensions.

What is the difference between a tangent line and a 2d linearization?

It is exactly the same concept, except brought into R³. Just as a 2-d linearization is a predictive equation based on a tangent line which is used to approximate the value of a function, a 3-d linearization is a predictive equation based on a tangent plane which is used to approximate a function.

 

[newyorkcity]

Looking back at the text, I just missed the relationship between (a,b), (x0,y0), and (x,y). To clarify, the 'Tangent Plane' equation is used to find the tangent plane at a point P(x0,y0,z0). The 'Linearization' equation yields the linear approximation of f(x,y) at (a,b). Thanks for your help.

 

[HallsofIvy]

Yes, just as the "linearization" of y= f(x) gives the tangent line to the curve, so the "linearization" of z= f(x,y) gives the tangent plane to the surface. The only difference is that when you talk about the "linearization" you are thinking of the functions and when you talk about the tangent plane you are thinking about their graphs.

 

[HallsofIvy]

This same question was asked in "Homework: Caculus and Beyond" so I am combining the two threads. newyorkcity, do NOT post the same question more than once.