Calculus of
Several Real Variables

This page is a sub-page of the page on Calculus.

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The sub-pages of this page are:

Functions of Several Real Variables
Partial and Directional Derivatives
Differentiation and Affine Approximation
Gradients
The Chain Rule in Several Real Variables
Taylor Expansion in in Several Real Variables
Riemann Integration in Several Real Variables
Partial Differential Equations
Vector Analysis
Differentials

Vektoranalys (in Swedish)

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Related KMR-pages:

Calculus of One Real Variable
Linear Algebra
• Matrix Algebra

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Books:

Differential and Integral Calculus by Richard Courant.

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Other related sources of information:

The right way to think about derivatives and integrals

Lamar University / Department of Mathematics / Class notes

Paul’s online notes
Paul’s Online Notes: Calculus II

Mathematical Analysis
Methods of Mathematical Physics, by Richard Courant and David Hilbert
Brilliant – Math and science done right
Multivariable calculus, Khan Academy on YouTube

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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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Several Real Variables

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