What is Geometric algebra: Definition and 41 Discussions
In mathematics, the geometric algebra (GA) of a vector space with a quadratic form (usually the Euclidean metric or the Lorentz metric) is an algebra over a field, the Clifford algebra of a vector space with a quadratic form with its multiplication operation called the geometric product. The algebra elements are called multivectors, which contains both the scalars
F
{\displaystyle F}
and the vector space
V
{\displaystyle V}
.
Clifford's contribution was to define a new product, the geometric product, that unified the Grassmann and Hamilton algebras into a single structure. Adding the dual of the Grassmann exterior product (the "meet") allows the use of the Grassmann–Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra (CGA) providing a framework for classical geometries. In practice, these and several derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations.
The scalars and vectors have their usual interpretation, and make up distinct subspaces of a GA. Bivectors provide a more natural representation of the pseudovector quantities in vector algebra such as oriented area, oriented angle of rotation, torque, angular momentum, electromagnetic field and the Poynting vector. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of
V
{\displaystyle V}
and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike vector algebra, a GA naturally accommodates any number of dimensions and any quadratic form such as in relativity.
Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has also found use as a computational tool in computer graphics and robotics.
The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics.
For the electrical resistance ##R## of an ideal wire, we all know the formula ##R=\rho * \frac{l}{A}##. However this is only valid for a cylinder with constant cross sectional area ##A##.
In a cone the cross section area is reduced over its height (or length ##l##). What is a good general...
Let the (multi-vector valued) “inner product” between a j-vector U and a k-vector B be defined as the (k-j) grade part of the geometric product UB, (a.k.a. “left contraction”) that is,
$$U\cdot B := <UB>_{k-j}$$
(0 when j > k) as is done in Alan Macdonald’s book “Linear and Geometric Algebra.”...
Homework Statement
I was working out problem 4, chapter 3 of Introduction to Electrodynamics by Griffiths:
a) Show that the average electric field over a spherical surface, due to charges outside the sphere, is the same as the field at the centre.
b) What is the average due to charges inside...
Hi, I am trying to learn Geometric Algebra by going through the book "New Foundations for Classical Mechanics" by David Hestenes.
I was reading the part about reduction formula (shown below) but couldn't get the result the shown in the book.
Can someone show me how iterating (1.15) gives the...
Homework Statement
Prove that ##\vec {a} \cdot (\vec {b} \wedge \vec {C_r}) = \vec {a} \cdot \vec {b} \vec {C_r} - \vec {b} \wedge (\vec {a} \cdot \vec {C_r})##.
Note that ##\vec {a}## is a vector, ##\vec {b}## is a vector, and ##\vec {C_r}## is an r-blade with ##r > 0##.
Also, the dot...
I have a simple paper with parallel grid lines and I know the height of the observer from the ground level.
Now when I snap the picture with my camera, the effect i get is what is seen in the image below. Even if the lines are parallel, the perception is that they converge in the middle.
When...
Hello! I am reading so very introductory stuff on geometric algebra and at a point the author says that, as a rule for calculation geometric products, we have that ##e_{12..n}=e_1\wedge e_2 \wedge ...\wedge e_n = e_1e_2...e_n##, with ##e_i## the orthonormal basis of an n-dimensional space, and I...
I am investigating the mathematical properties of a vector-product. I am wondering if it might be old-hat in GA (which is new to me)?
I am using the working-title "spin-product" for a vector-product that combines RANDOM rotation-only of a direction-vector [a unit 1-vector; say...
I want to understand what changing coordinate system means for hands of clock. Let's say the clock only has hour and minute hand. It can move let's say just in the upper 180 deg. of the clock (as shown in the figure). The area between the two hands is V1, and the rest is V2. Depending on the...
Hi all,
I'm reading a paragraph from "Geometric Algebra for Physicists" - Chris Doran, Anthony Lasenby. I'm quite interested in applying GA to QM but I've got to a stage where I am not following part of the chapter and am wondering if someone can shed a little light for me.
The part...
Homework Statement
Let S be a sphere with the equation ##(x-2)^2+y^2+z^2=2 ## and let p a line which satisfies the condition ## p \in (\Pi \cap \Sigma) ## where ##\Pi## and ##\Sigma## are planes with equations:
##\Pi :x+z=2##
##\Sigma: 5x-2z=3##
a) Show that S and p have exactly one common...
If we seek a bijection $$\wedge^p V \to \wedge^{n-p} V$$ for some inner product space ##V##, we might think of starting with the unit ##n##-vector and removing dimensions associated with the original vector in ##\wedge^p V ##. Might this be expressed as a sequence of steps by some binary...
I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby.
They claim in chapter 4 that the geometric product ab between two vectors a and b is defined according to the axioms
i) associativity: (ab)c = a(bc) = abc
ii) distributive over addition: a(b+c) = ab+ac
iii) The...
I'm learning geometric algebra. There is a very simple statement which I think is wrong. But it must be right, because all the experts say so. Arrg!
The only properties used are
1a = a1 = a
aa = 1
if b<>a then ab = -baTheir claim is that abcdabcd = -1.
Let's see:
aa = 1
abab = -abba =...
On this picture we see a octagonal dome. I am trying to calculate the volume of this object by integral calculus but I can't find a way. How would you calculate this?
https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202015-12-17%20kl.%2002.14.48.png [Broken]
I am majoring in...
Hi,
I have been wanting to do this for a while but not too sure how to go about it. I have the following geometric algebra
\lbrace\mathbf{e}_{i}\rbrace_{i=0}^{3} which satisfy the following relations: \mathbf{e}_{i}\mathbf{e}_{j}=-\mathbf{e}_{j}\mathbf{e}_{i} and...
I was given the following as a proof that the inertial tensor was symmetric. I won't write the tensor itself but I will write the form of it below in the proof. I am confused about the steps taken in the proof. It involves grade projections.
A \cdot (x \wedge (x \cdot B)) = \langle Ax(x...
Hi, I want to calculate a rotation of a vector GA style with this formula e^{-B \frac{\pi}{2}}(2e_{1}+3e_{2}+e_{3})e^{B\frac{\pi}{2}}. Now since no book/pdf on GA exists where a calculation is explicitly done with numbers, I wounder how to calculate this. Should I substitude e^{-B...
For those unaware, geometric algebra is a mathematical language that generalizes and simplifies a lot of the tools physicists work with (vectors, complex numbers, tensors, etc). I found a neat example in classical mechanics to illustrate its power. Simply, a charge in a constant magnetic field...
I was curious if anyone here ever studied Geometric Algebra? It seems not so mainstream and fairly new and I feel intrigued by the subject but I don't want to get in over my head. Just browsing through the table of contents of some books has a lot of unfamiliar terms to me.
Here was a...
I've seen a number of books and articles touting Geometric Algebra as an important new area of math that will have large application to physics. Is there anything to these claims? Is it worth studying for a physics student?
Hello,
I have the following equation in x and y: xy - \sqrt{(x^2+a^2)(y^2+c^2)} = -\frac{1}{a^2}-\frac{1}{c^2} where the quantities a2 and c2 are given real constants, and I have to find real values for x, and y such that the equation above is always satisfied.
Actually, I know that the...
Hello,
I have recently started to study some Geometric Algebra.
I was wondering how should I interpret complex-vectors in \mathcal{C}^n in the framework of Geometric Algebra.
I understand already that a complex-scalar should be interpreted as an entity of the kind:
z = x + y (\textbf{e}_1...
Recently I discovered geometric algebra which looks very exciting. I was wondering if there is any connection between geometric algebra and differential forms?
I see that different research groups recommend the use of differential forms (http://www.ee.byu.edu/forms/forms-home.html" [Broken])...
I'm trying to study geometric algebra using Artin's book and am having some difficulty with what degenerate symmetric bilinear forms would be like. Does someone know of an example and brief explanation. Also, the opposite being "nondegenerate nonsymmetric bilinear form" would help me out. If I...
[SOLVED] Geometric Algebra: Signs of electromagnetic field tensor components?
Here's a question that may look like an E&M question, but is really just a geometric algebra question. In particular, I've got a sign off by 1 somewhere I think and I wonder if somebody can spot it.
PF isn't...
[SOLVED] geometric algebra: longitude and latitude rotor ordering?
Was playing around with what is probably traditionally a spherical trig type problem using geometric algebra (locate satellite position using angle measurements from two well separated points). Origin of the problem was just me...
Anybody know a good approach to factor a wedge product (blade) into a set of vectors? Loosely, I'd describe the prodlem as finding a basis for the hypervolume that the wedge product "spans".
Example to illustrate the question, taking a grade 2 blade, suppose one had something like:
A =...
My differential forms book (Flanders/Dover) defines an inner product on wedge products for vectors that have a defined inner product, and uses that to define the hodge dual. That wedge inner product definition was a determinant of inner products.
I don't actually have that book on me right...
Given the parametric representation of two planes, through points P and Q respectively
x = P + \alpha u + \beta v
y = Q + a w + b z
Or, alternately, with u \wedge v = A, and w \wedge z = B
x \wedge A = P \wedge A
y \wedge B = Q \wedge B
It's easy enough to find...
geometric algebra cross product
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Homework Statement [/b]
my text (Geometric Algebra for Physicists, by Doran and Lasenby), p. 69, deals with rotating frame {fsubk} (I assume in 3D)
d/dt (fsubk) = omega X...
Homework Statement [/b]
my text (Geometric Algebra for Physicists, by Doran and Lasenby), p. 69, deals with rotating frame {fsubk} (I assume in 3D)
d/dt (fsubk) = omega X fsubk omega being angular velocity
then
omega X fsubk = (-I omega) dot fsubk = fsubk dot (I omega), where...
This example appears in a set of notes entitled Geometric Algebra.
I cannot follow the first half of the example. Is the reasoning incorrect.
Thanks. Matheinste.
I've been studying geometric algebra of the form promoted by David Hestenes, but I'm having trouble with the very basics.
Most GA books, in fact, all GA books, begin as follows.
For two vectors \mathbf{a}\mathbf{b}, they define the symmetrical inner product...
I've been looking into Geometric Algebra approaches to linear transformations and have found it to be MUCH nicer than the conventional matrix approaches for certain kinds of transformations. Moreover, I find it much more intuitive, particularly in its way of dealing with complex numbers.
For...
Why don't we discuss the Geometric algebra and how it differs from other Clifford algebras?
For introduction, here's Hestenes' home page on Geometric calculus:
http://modelingnts.la.asu.edu/
This is an easy reading introduction:
(1) GA seamlessly integrates the properties of vectors...
Consider 3D geometric algebra. Let all points on a line be given by the parametrization x=tu+y, in which the parameter runs from minus infinity to plus infinity.
a. Show that for all points on the line we have
x(wedge)u=y(wedge)u.
b. Show that the vector d pointing from the...
can anyone explain how commutators act on tri-vectors (in orthonormal conditions)?
on bi-vectors i know that it ends up to be a bivector again,
but with tri-vectors it vanishes if its lineraly dependent.
what about the case if its not linearly dependent,
does that mean it remains a...