1.2. Polar Form
... Many physical quantities are a function of more than one variable (e.g. the pressure of a gas depends upon both temperature and volume, a magnetic field may be a function of the three spatial co-ordinates (x,y,z) and time (t)). Hence when differentiating a function there is usually a choice of which ...
... Many physical quantities are a function of more than one variable (e.g. the pressure of a gas depends upon both temperature and volume, a magnetic field may be a function of the three spatial co-ordinates (x,y,z) and time (t)). Hence when differentiating a function there is usually a choice of which ...
16. Subspaces and Spanning Sets Subspaces
... But we know that any two vector define a plane. In this case, the vectors in U define the xy-plane in R3 . We can consider the xy-plane as the set of all vectors that arise as a linear combination of the two vectors in U . Call this set of all linear combinations the span of U : ...
... But we know that any two vector define a plane. In this case, the vectors in U define the xy-plane in R3 . We can consider the xy-plane as the set of all vectors that arise as a linear combination of the two vectors in U . Call this set of all linear combinations the span of U : ...
Cartesian tensor
In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is through an orthogonal transformation.The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product.Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system.