For vector equations, sometimes the defining quantity is in a cross or dot product and cannot be solved for explicitly as a vector, but the components can.
12.
In the special case of a single particle with no electric charge and no spin, the orbital angular momentum operator can be written in the position basis as a single vector equation:
13.
The equations above thus represent respectively conservation of mass ( 1 scalar equation ) and momentum ( 1 vector equation containing N scalar components, where N is the physical dimension of the space of interest ).
14.
With this new dependent vector variable, the Navier-Stokes equation ( with curl taken as above ) becomes a single fourth order vector equation, no longer containing the unknown pressure variable and no longer dependent on a separate mass continuity equation:
15.
Note that if the cross differentiation is left out, the result is a third order vector equation containing an unknown vector field ( the gradient of pressure ) that may be determined from the same boundary conditions that one would apply to the fourth order equation above.
16.
The incompressible Navier-Stokes equation with mass continuity ( four equations in four unknowns ) can, in fact, be reduced to a single equation with a single dependent variable in 2D, or one vector equation in 3D . This is enabled by two vector calculus identities:
17.
Expressing the Navier Stokes vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space employed, and this is the case also for the first-order terms ( like the variation and convection ones ) also in non-cartesian orthogonal coordinate systems.
18.
:The two constraints are the two equations, one for the plane and one for the sphere, each equation will be one constraint, regardless of how many variables are in it ( as long as it isn't a vector equation, anyway-vector equations represent multiple scalar equations ).
19.
:The two constraints are the two equations, one for the plane and one for the sphere, each equation will be one constraint, regardless of how many variables are in it ( as long as it isn't a vector equation, anyway-vector equations represent multiple scalar equations ).
20.
The first of these equations express the Newton's law and is the equivalent of the vector equation f = m a ( force equal mass times acceleration ) plus t = J \ dot { \ omega } + \ omega \ times J \ omega ( angular acceleration in function of inertia and angular velocity ); the second equation permits the evaluation of the linear and angular momentum when velocity and inertia are known.
