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:
• Multivariable Calculus Theory, by Serpentine Integral on YouTube.
• Understanding Lagrange Multipliers Visually, by by Serpentine Integral on YouTube.
• 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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