# Calculus of Several Real Variables

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Vektoranalys (in Swedish)

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Related KMR-pages:

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Books:

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Other related sources of information:

Mathematical Analysis
Methods of Mathematical Physics, by Richard Courant and David Hilbert
Brilliant – Math and science done right

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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What they won’t teach you in calculus (Steven Strogatz on YouTube):

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Functions of several real variables:

A function $f$ from $\mathbb{R}^2$ to $\mathbb{R}$ can be described by:

${{\mathbb{R}^2 \, \stackrel {f} {\longrightarrow} \, \mathbb{R} \:}\atop {\: (x,y) \, \longmapsto \, f(x,y) } } {\,}$.

The differential $\, df \,$ of the function $f$ at the point $(a,b) \in \mathbb{R}^2$ is given by:

$\, df_{(a,b)} = \frac{\partial f}{\partial x}_{(a,b)} dx + \frac{\partial f}{\partial y}_{(a,b)} dy$.

The equation of the level curve ($\, f = \text{constant} \,$) of the function $\, f \,$ at the point $\, (a,b) \,$
is given by:

$\, f(x,y)=f(a,b)$.

The equation of the tangent to the level curve of the function $\, f \,$ at the point $\, (a,b) \,$ is given by:

$\, \frac{\partial f}{\partial x}_{(a,b)} (x-a) + \frac{\partial f}{\partial y}_{(a,b)} (y-b) = 0$.

The normal to this tangent at the point $\, (a, b) \,$ is called the gradient of the function $\, f \,$ at the point $\, (a,b) \,$. It is given by the vector

$\, (\frac{\partial f}{\partial x}_{(a,b)}, \frac{\partial f}{\partial y}_{(a,b)})$.

NOTE: In three dimensions the level curve is a level surface, and the tangent line is a tangent plane. The gradient is still perpendicular to the tangent plane at the point of tangency. The relation between the gradients in two and three dimensions is visualized here.

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