Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Vector calculus identities
Summary
The following are important identities involving derivatives and integrals in vector calculus.
Gradient
For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables , also called a scalar field, the gradient is the vector field:
where are orthogonal unit vectors in arbitrary directions.
As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change.
For a vector field , also called a tensor field of order 1, the gradient or total derivative is the n × n Jacobian matrix:
For a tensor field of any order k, the gradient is a tensor field of order k + 1.
For a tensor field of order k > 0, the tensor field of order k + 1 is defined by the recursive relation
where is an arbitrary constant vector.
Divergence
In Cartesian coordinates, the divergence of a continuously differentiable vector field is the scalar-valued function:
As the name implies the divergence is a measure of how much vectors are diverging.
The divergence of a tensor field of non-zero order k is written as , a contraction to a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity,
where is the directional derivative in the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors,
For a tensor field of order k > 1, the tensor field of order k − 1 is defined by the recursive relation
where is an arbitrary constant vector.
Curl (mathematics)
In Cartesian coordinates, for the curl is the vector field:
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.
As the name implies the curl is a measure of how much nearby vectors tend in a circular direction.
In Einstein notation, the vector field has curl given by:
where = ±1 or 0 is the Levi-Civita parity symbol.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.