Foundations — Differentiability in multiple variables

Before you can read the parent note, you need every piece of its language built from the ground up. We go symbol by symbol, each one earning its place with a plain-words meaning and a picture, in an order where every step leans on the one before it.

1. Points and the plane: $(x,y)$ and ${\mathbb{R}}^2$

• Plain words: $\mathbb{R}$ is the number line; ${\mathbb{R}}^2$ is two number lines crossed at right angles.
• Picture: a floor tiled with a grid; every crossing is one point $(x,y)$.
• Why the topic needs it: our function lives above this floor. The input is always a point on the floor.

2. The function $f:{\mathbb{R}}^2\to \mathbb{R}$ and its graph (a surface)

• Plain words: feed in a spot on the floor, get back how high the ground is there.
• Picture: over every floor-point you stack a vertical pole of length $f(x,y)$; the tips of all the poles form a bumpy surface — the graph of $f$.
• Why the topic needs it: "differentiability" is a question about this surface — is it smooth, or secretly broken?

The height is often called $z$, so $z=f(x,y)$.

See Continuity in Multiple Variables for what "no sudden jumps in the surface" means precisely.

3. The limit arrow $\lim$ and $h\to 0$

• Plain words: "where is this heading as the step gets tiny?"
• Picture: a slider $h$ sliding down to $0$; you watch the output number and read off where it lands.
• Why this tool and not just plugging in $0$? Many of our expressions are $\frac{0}{0}$ when you set $h=0$ outright — meaningless. The limit lets us ask the trend instead of the forbidden division.

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

To talk about "how steep" we first recall the 1-D idea of slope, then aim it along one axis.

Now hold $y$ fixed and only walk East (change $x$):

• Picture: slice the surface with a vertical wall running East–West through $(a,b)$. The cut edge is a curve; $f_x$ is its slope.
• $f_y$ is the same idea, wiggling $y$ (walking North), slicing with a North–South wall.
• Why the topic needs it: these are the only two slopes partials give you — and the parent note's whole warning is that two slopes are not enough.

Full treatment: Partial Derivatives.

5. Vectors: $\mathbf{h}$, $\mathbf{a}$, and the bold-face convention

• Plain words: $\mathbf{h}$ is "how far and which way I stepped off the base point".
• Picture: an amber arrow starting at $\mathbf{a}$ and pointing to the new spot $\mathbf{a}+\mathbf{h}$.
• Why the topic needs it: differentiability must hold for every step $\mathbf{h}$ — not just East and North, but every diagonal. The vector is how we say "every direction at once".

6. Length of a step: $\Vert \mathbf{h}\Vert$

• Plain words: the ruler-distance from $\mathbf{a}$ to $\mathbf{a}+\mathbf{h}$.
• Picture: the arrow $\mathbf{h}$ is the hypotenuse of a right triangle with legs $h_1$ and $h_2$.
• Why this tool? The definition of differentiability divides the error by $\Vert \mathbf{h}\Vert$. To divide by "how far you stepped" we must first measure that distance — that's exactly what the norm does.

7. The gradient $\nabla f$ and the dot product $\cdot$

• Plain words: $\nabla f\cdot \mathbf{h}$ = "predicted height-change if you take step $\mathbf{h}$" = (East-slope × East-part of step) + (North-slope × North-part of step).
• Picture: the tilted flat board (tangent plane) rising by exactly this amount over the step $\mathbf{h}$.
• Why the topic needs it: this dot product is the tilted plane's height-change. The whole game is: does the real surface change by this amount, plus something negligibly small?

More: Gradient and Directional Derivatives.

8. "Negligibly small": the little-o, $o(\Vert \mathbf{h}\Vert )$

• Plain words: not just "the error goes to zero", but "the error is puny compared to the distance you moved".
• Picture: two curves diving to zero as $\Vert \mathbf{h}\Vert \to 0$ — the error dives below the straight line $\Vert \mathbf{h}\Vert$ and stays there.
• Why this tool and not just $E\to 0$? Every continuous function has $E\to 0$. That's too weak — it can't tell a genuine tangent plane from any plane grazing the point. Dividing by $\Vert \mathbf{h}\Vert$ is the sharper test that pins down tangency.

9. Putting the sentence together

With every symbol earned, the parent's master formula now reads in plain English:

$f(\mathbf{a}+\mathbf{h})=\underset{\text{start height}}{\underbrace{f(\mathbf{a})}}+\underset{\text{tilted-plane change}}{\underbrace{\nabla f(\mathbf{a})\cdot \mathbf{h}}}+\underset{\text{negligible error}}{\underbrace{o(\Vert \mathbf{h}\Vert )}}$

"New height = old height + what the flat tilted board predicts + a leftover so tiny it doesn't count." That leftover being tiny enough — smaller than the step length — is the whole definition of differentiability.

Downstream this feeds Tangent Planes and Linear Approximation, the Chain Rule (Multivariable), the Total Derivative / Jacobian, and the notion of C1 and Smooth Functions.

Prerequisite map

Equipment checklist