# Simplicial calculus with Geometric Algebra

@inproceedings{Sobczyk1992SimplicialCW, title={Simplicial calculus with Geometric Algebra}, author={G. Sobczyk}, year={1992} }

We construct geometric calculus on an oriented k-surface embedded in Euclidean space by utilizing the notion of an oriented k-surface as the limit set of a sequence of k-chains. This method provides insight into the relationship between the vector derivative, and the Fundamental Theorem of Calculus and Residue Theorem. It should be of practical value in numerical finite difference calculations with integral and differential equations in Clifford algebra.

#### 40 Citations

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Sobczyk's simplicial calculus does not have a proper foundation

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The pseudoscalars in Garret Sobczyk's paper \emph{Simplicial Calculus with Geometric Algebra} are not well defined. Therefore his calculus does not have a proper foundation.

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#### References

SHOWING 1-10 OF 12 REFERENCES

Analysis, manifolds, and physics

- Physics
- 1977

Contents. Preface. Preface to the second edition. Preface. Contents. Conventions. I. Review of fundamental notions of analysis. II. Differential calculus on banach spaces. III. Differentiable… Expand

Generalized Functions N.Y

- Generalized Functions N.Y
- 1984

Analysis, Manifolds and Physics N.Y

- Analysis, Manifolds and Physics N.Y
- 1977

N.Y

- N.Y
- 1969

N.Y

- N.Y
- 1969

Topological Geometry

- Cambridge Univ. Press, N.Y.
- 1969

Multivector Calculus

- J. Math. Anal. & Appl
- 1968

Multivector Functions

- J. Math. Anal. & Appl
- 1968