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In Spivak's "Calculus on Manifolds" he introduces partial derivatives as follows.

If $g(x)=f(a^1,\dots,x,\dots,a^n)$, then $D_if(a)=g'(a^i)$. This means that $D_if(a)$ s the slope of the tangent line at $(a,f(a))$ to the curve obtained by intersecting the graph of $f$ with the plane $x^j=a^j$, $j\ne i$.

I've interpreted this as meaning that the $i$th partial derivative is simply the $i$th column of the matrix form of the total derivative of the function $f$. However, I am not so sure how he draws the conclusion that it is the slope of the tangent line to the curve obtained by intersecting the graph of $f$ with the plane $x^j=a^j$, since if we consider a general case where $n>2$ there would be multiple (2) choices to create the curve by this method, which would correspondingly give different slopes of the tangent lines. Where is the error in my reasoning?

Redcrazyguy
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  • Briefly, fix every variable except $x^i$ ("for $j \neq i$") and consider the resulting function of one variable. By "plane" Spivak means a hyperplane, i.e., an affine subspace of dimension $n-1$. Please feel free to answer your own question if that clarifies! – Andrew D. Hwang Aug 30 '22 at 17:37
  • @AndrewD.Hwang so for example if it's a function of three variables, it would be the tangent of the curve resulting from the plane formed from keeping $x^2$ and $x^3$ constant? – Redcrazyguy Aug 30 '22 at 17:49
  • $D_1f$ would be, yes. – Andrew D. Hwang Aug 30 '22 at 21:04

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