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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