Calculus of Several Real Variables

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


The sub-pages of this page are:

Functions of Several Real Variables
Partial and Directional Derivatives
Differentiation and Affine Approximation
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

Vektoranalys (in Swedish)


Related KMR-pages:

Calculus of One Real Variable
Linear Algebra
• Matrix Algebra



Differential and Integral Calculus by Richard Courant.


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


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


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


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.

11 thoughts on “Calculus of Several Real Variables

  1. Wow, superb blog layout! How long have you been blogging for? you make blogging look easy. The overall look of your website is wonderful, as well as the content!

  2. Hello there, just became aware of your blog through Google, and found that it is truly informative. I will be grateful if you continue this in future. A lot of people will benefit from your writing. Cheers!

  3. I needed to thank you for this great read!! I absolutely loved every little bit of it. I’ve got you bookmarked to look at new things you post.

  4. This site certainly has all the info I needed concerning this subject and didn’t know who to ask.

  5. whoah this blog is excellent i love studying your articles. Keep up the great work! You know, a lot of persons are hunting around for this info, you can aid them greatly.

  6. I just like the valuable info you supply in your articles. I will bookmark your blog and take a look at once more here regularly. I’m sure I’ll be told lots of new stuff proper right here! Best of luck for the following!

  7. Great work! This is the kind of information that should be shared across the internet. Shame on Google for now not positioning this post upper! Come on over and consult with my website . Thanks =)

  8. I’m impressed, I must say. Rarely do I come across a blog that’s equally educative and entertaining, and let me tell you, you’ve hit the nail on the head. The issue is something which too few people are speaking intelligently about. I’m very happy I came across this during my search for something regarding this.

  9. If some one wishes expert view on the topic of running a blog afterward i advise him/her to visit this website, Keep up the pleasant work.

  10. I’ve been browsing online more than three hours lately, yet I never discovered any interesting article like yours. It is beautiful worth enough for me. Personally, if all site owners and bloggers made good content material as you do, the net might be much more helpful than ever before.|

  11. These are in fact wonderful ideas in concerning blogging. You have touched some fastidious points here. Any way keep up wrinting.

Leave a Reply

Your email address will not be published. Required fields are marked *