Abstract
We recall that one-dimensional definite integral introduced in Sect. 4.1 is related to functions of a single variable. Here we shall generalise the idea of integration for functions of more than one variable. We start from the simplest case—integration of functions of two variables.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
After all, in Sect. 4.2 we did not actually made use of the fact that what we sum corresponds to a one-dimensional integral; we simply considered the limit of an integral sum.
- 2.
Note that on the left and right of the inequality prior to integration we had constant functions m kj and M kj which do not depend on x, so that integration simply corresponds to multiplying by \(\Delta y_{j}\) in either part.
- 3.
We will use the following result: \(\int \sqrt{a^{2 } - t^{2}}dt = \frac{1} {2}\left [t\sqrt{a^{2 } - t^{2}} + a^{2}\arcsin \frac{t} {a}\right ] + C\), see Eq. ( 4.60).
- 4.
Named after Carl Gustav Jacob Jacobi.
- 5.
The corresponding indefinite integral cannot be represented by a combination of elementary analytical functions.
- 6.
Once again, we need to use the integral of Eq. ( 4.60).
- 7.
Although in this particular case, it does not affect the final zero result!
- 8.
The strip was also discovered independently by Johann Benedict Listing.
- 9.
Note that \(\mathbf{N} \sim \mathbf{r}\) as one would intuitively expect for the normal to a sphere!
- 10.
Note that the Laplace equations here and in electrostatics differ by an unimportant factor of \(-4\pi\).
- 11.
Note that it is customary in electrostatics to choose the minus sign when defining the potential.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Kantorovich, L. (2016). Functions of Many Variables: Integration. In: Mathematics for Natural Scientists. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2785-2_6
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2785-2_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2784-5
Online ISBN: 978-1-4939-2785-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)