The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces such as gravity. All non-relativistic balance equations, such as the Navier—Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation. By expressing the deviatoric shear stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier—Stokes equations below. Though the flow may be steady time-independentthe fluid decelerates as it moves down the diverging duct assuming incompressible or subsonic compressible flowhence there is an acceleration happening over position.
Definition[ edit ] Let K be a field such as the real numbersV be a vector space over K, and let W be a subset of V. Then W is a subspace if: The zero vector0, is in W. If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W. Examples[ edit ] Example I: Let the field K be the set R of real numbersand let the vector space V be the real coordinate space R3.
Take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V. Thus, cu is an element of W too. Let the field be R again, but now let the vector space be the Cartesian plane R2.
Then W is a subspace of R2. In general, any subset of the real coordinate space Rn that is defined by a system of homogeneous linear equations will yield a subspace. Geometrically, these subspaces are points, lines, planes, and so on, that pass through the point 0.
Examples related to calculus[ edit ] Example III: Let C R be the subset consisting of continuous functions.
Then C R is a subspace of RR. We know from calculus that the sum of continuous functions is continuous. Again, we know from calculus that the product of a continuous function and a number is continuous.
Keep the same field and vector space as before, but now consider the set Diff R of all differentiable functions. The same sort of argument as before shows that this is a subspace too. Examples that extend these themes are common in functional analysis. Properties of subspaces[ edit ] A way to characterize subspaces is that they are closed under linear combinations.
That is, a nonempty set W is a subspace if and only if every linear combination of finitely many elements of W also belongs to W. Conditions 2 and 3 for a subspace are simply the most basic kinds of linear combinations.
In a topological vector space X, a subspace W need not be closed in general, but a finite-dimensional subspace is always closed. Descriptions[ edit ] Descriptions of subspaces include the solution set to a homogeneous system of linear equationsthe subset of Euclidean space described by a system of homogeneous linear parametric equationsthe span of a collection of vectors, and the null spacecolumn spaceand row space of a matrix.
Geometrically especially, over the field of real numbers and its subfieldsa subspace is a flat in an n-space that passes through the origin. A natural description of an 1-subspace is the scalar multiplication of one non- zero vector v to all possible scalar values.A System of Linear Equations is when we have two or more linear equations working together.
/*A C program is executed as if it is a function called by the Operating System, the Operating System can and does pass parameters to the program.
which clearly has no solution. The system is inconsistent.
Notes. If a matrix is carried to row-echelon form by means of elementary row operations, the number of leading 1's in the resulting matrix is called the rank $r$ of the original matrix.
Suppose that a system of linear equations in $n$ variables has a solution. A Diophantine equation is a polynomial equation whose solutions are restricted to integers.
These types of equations are named after the ancient Greek mathematician Diophantus. A linear Diophantine equation is a first-degree equation of this type. Diophantine equations are important when a problem requires a solution in whole amounts.
The study of problems that require integer solutions is. Page 1 of 2 Solving Systems Using Inverse Matrices SOLUTION OF A LINEAR SYSTEM Let AX= Brepresent a system of linear equations.
If the determinant of Ais nonzero, then the linear system has exactly one solution, which is X= Aº1B. Solving a Linear System Use matrices to solve the linear system in Example 1.
kcc1 Count to by ones and by tens. kcc2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
kcc3 Write numbers from 0 to Represent a number of objects with a written numeral (with 0 representing a count of no objects). kcc4a When counting objects, say the number names in the standard order, pairing each object with one and only.