Vector fields in cylindrical coordinates

Vectors are defined in cylindrical coordinates by (ρ,φ,z), where This article describes some of the common coordinate systems that appear in elementary mathematics. ...

• ρ is the length of the vector projected onto the X-Y-plane,
• φ is the angle of the projected vector with the positive X-axis (0 ≤ φ < 2π),
• z is the regular z-coordinate.

(ρ,φ,z) is given in cartesian coordinates by: Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

or inversely by:

Any vector field can be written in terms of the unit vectors as: Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

The cylindrical unit vectors are related to the cartesian unit vectors by:

• Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

Time derivative of a vector field in cylindrical coordinates

To find out how the vector field A changes in time we calculate the time derivatives. In cartesian coordinates this is simply:

However, in cylindrical coordinates this becomes:

We need the time derivatives of the unit vectors. They are given by:

So the time derivative simplifies to:

Gradient, divergence, curl, and laplacian in cylindrical coordinates

The specification of gradient, divergence, curl, and laplacian in cylindrical coordinates can be found in the article Nabla in cylindrical and spherical coordinates. In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... This article is about the cURL command line tool. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... This is a list of some vector calculus formulae of general use in working with standard coordinate systems. ...

Vector fields in spherical coordinates

Vectors are defined in spherical coordinates by (r,θ,φ), where This article describes some of the common coordinate systems that appear in elementary mathematics. ...

• r is the length of the vector,
• θ is the angle with the positive Z-axis (0 <= θ <= π),
• φ is the angle with the X-Z-plane (0 <= φ < 2π).

(r,θ,φ) is given in cartesian coordinates by: Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

or inversely by:

Any vector field can be written in terms of the unit vectors as:

The spherical unit vectors are related to the cartesian unit vectors by:

• Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

Time derivative of a vector field in spherical coordinates

To find out how the vector field A changes in time we calculate the time derivatives. In cartesian coordinates this is simply:

However, in spherical coordinates this becomes:

We need the time derivatives of the unit vectors. They are given by:

So the time derivative becomes:

Gradient, divergence, curl, and laplacian in spherical coordinates

The specification of gradient, divergence, curl, and laplacian in spherical coordinates can be found in the article Nabla in cylindrical and spherical coordinates. In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... This article is about the cURL command line tool. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... This is a list of some vector calculus formulae of general use in working with standard coordinate systems. ...