Foundations — Tangent planes and linear approximations to surfaces

This page assumes nothing. Before you can read the parent note Tangent Planes, every symbol it uses is built here from the ground up, in an order where each one leans only on the ones before it.

0. The picture we live inside: 3D space and $(x,y,z)$

Everything happens in three-dimensional space. We label a point by three numbers.

Why the topic needs it: a surface is a shape floating in this space, and the tangent plane is another shape in the same space. Without axes we can't say where anything is.

1. A function of two variables: $f(x,y)$

Why the topic needs it: the surface is the function. The whole game is approximating $f$ near one spot.

2. The surface $z=f(x,y)$

The bowl in the figure is $z=x^2+y^2$. Directly above the ground spot $(a,b)$ sits the surface point $(a,b,f(a,b))$ — the yellow dot. That dot is where our flat sheet will touch.

3. The base point $(a,b)$ and the offsets $x-a$, $y-b$

Why the topic needs it: "linear approximation" means height at base plus slope $\times$ step. The step is precisely $(x-a,y-b)$.

4. Slope in one direction: the partial derivative $f_x$, $f_y$

Before two dimensions, recall the one-dimensional idea from Tangent line and linear approximation (single variable): the slope of a curve $y=f(x)$ is "rise over run" of its tangent line — how fast height changes as you step right.

Why the topic needs it: the tangent plane must tilt exactly like the surface. Its two tilts are $f_x$ and $f_y$. Full machinery lives in Partial derivatives.

5. A plane written as height + tilt

Read the symbols: at the base point $x-a=0$ and $y-b=0$, so $z=z_0$ — the sheet passes through height $z_0$. Step one unit right ($x-a=1$) and $z$ rises by $A$: that's why $A$ is the rightward slope.

Why the topic needs it: this is the raw shape of the answer. The parent note fills in $z_0=f(a,b)$, $A=f_x$, $B=f_y$ to make it the tangent plane.

6. The linear approximation $L(x,y)$

Once we plug $z_0=f(a,b)$, $A=f_x(a,b)$, $B=f_y(a,b)$ into the plane of section 5, the right-hand side gets its own name.

Why the topic needs it: every later formula — the $\approx$ statement, the error, the differentiability limit — is written in terms of $L$. Having a single letter for "the flat-sheet height" keeps those formulas short.

7. Approximately equal: the symbol $\approx$

Why the topic needs it: the whole point of a linear approximation is that it's a good guess, not an exact truth. The gap is the error, studied via Differentiability of multivariable functions.

8. Change symbols: $\Delta z$, $dz$, $dx$, $dy$

Why the topic needs it: differentials $dz$ are how the topic does fast error estimates (like the rectangle-area example).

9. The limit and "shrinks faster": the differentiability test

Why the topic needs it: this is the fine print guaranteeing the plane truly "hugs" the surface — the subject of Differentiability of multivariable functions.

10. Vectors and the normal $\mathbf{n}=⟨f_x,f_y,-1⟩$

Why the topic needs it: it gives a second, elegant way to write the plane, and it connects to The gradient vector and Directional derivatives.

How these feed the topic

Equipment checklist

Can you answer each without peeking? Reveal to check.

Connections

• Tangent line and linear approximation (single variable) — the 1D slope idea we upgraded.
• Partial derivatives — the machinery behind $f_x,f_y$.
• The gradient vector — packages the partials into the normal.
• Differentiability of multivariable functions — the limit test made precise.
• The chain rule (multivariable) — next tool built on this foundation.
• Directional derivatives — slope in any direction, living in the tangent plane.

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