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I am reading out of Marsdens Vector Calculus and the text gives the same equation under two different headings, namely Linear or Affine Approximations and Tangent Plane to a Surface. The equation is the following at given points $(x_0, y_0)$: $$z= f(x_0 ,y_0) +[\frac{\partial f}{\partial x} (x_0,y_0) ](x - x_0)+[\frac{\partial f}{\partial y} (x_0,y_0) ](y - y_0)$$

Question 1 Can we generalize this equation to an n-dimensional object which has a tangent (n-1) dimensional object at point $(x_1, x_2, \ldots x_{n-1})$?? Or is the case that this equation of the form

$$z= f(x_1,x_2, \ldots x_{n-1}) +[\frac{\partial f}{\partial x_1} (x_1,x_2, \ldots x_{n-1} ) ](x-x_1)+ \cdots + [\frac{\partial f}{\partial x_{n-1}} (x_1,x_2, \ldots x_{n-1})](x- x_{n-1})$$ will represent the tangent plane of an n-dimensional object at point $(x_1, x_2, \ldots x_{n-1})$ in $\mathbb R^n$?

Question2: Can someone prove either of these statements if they are true or suggest where I might find it?

Question 3: Is the equation of affine approximation and the tangent plane always the same equation as dimensions increase or does the affine approximation become expressed by a different equation than the tangent plane equation(or the tangent $n-1$ dimensional-object equation which ever is the case with the equation of this form)??

Thanks in advance, and I apologize for any errors in questions if there be, I'm working off a phone and it is incredibly frustrating.

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  • $\begingroup$ Sorry I had made a huge mistake I just corrected. $\endgroup$ – Red Nov 5 '16 at 18:57
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    $\begingroup$ This is sort of a collection of yes/no questions. 1. yes you get the tangent (n-1) dimensional object, 2. yes you can find a proof in an advanced multivariable calculus text, 3. yes affine approximations are the same as the tangent (n-1) dimensional objects $\endgroup$ – Mark S. Nov 5 '16 at 19:46
  • $\begingroup$ @Mark S. The text I'm using, that is, Vector Calculus by Tromba and Marsden does not have it. Do you know of common book that does?? And thanks for the answers. $\endgroup$ – Red Nov 5 '16 at 20:03
  • $\begingroup$ If I recall correctly, I believe Marsden and Tromba covers Taylor's Theorem. This would basically be the degree 1 case. $\endgroup$ – Mark S. Nov 5 '16 at 20:42
  • $\begingroup$ @Mark S. Yes it does and you just gave me a aha'' $\endgroup$ – Red Nov 5 '16 at 20:51
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(Condensing my comments into an answer.)

The proof of multivariate Taylor's Theorem covers this as the degree 1 case is the affine approximation by an $(n-1)$-dimensional object. As multivariate Taylor's Theorem is a generalization of the univariate one, the guess of "tangent plane formula but with more terms corresponding to more input variables" is a good one.

This affine approximation is often called the "tangent hyperplane (line/plane/space/hyperspace)". Be careful, in the field of differential geometry, there is a slightly different (related) definition of "tangent space".

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