"If you got the eye of a mathematician you will see patterns in everything"
First-order Partial Differential EquationsLet u = u(q,.. ., 2,) be a function of n independent variables z1, ...,2 ., A Partial Differential Equation (PDE for short) is an equation that contains the independent variables q , ... , Xn, the dependent variable or the unknown function u and its partial derivatives up to some order. It has the form where F is a given function and u(Xj)= du / d X j , Uxixj = d2u/dxidxj... Uxi , Uxj are the partial derivatives of u. The order of a PDE is the order of the highest derivative which appears in the equation.
A set Ω in the n-dimensional Euclidean space Rn is called a domain if it is an open and connected set. A region is a set consisting of a domain plus, perhaps, some or all of its boundary points. We denote by C(R) the space of continuous functions in Ω and by Ck (a) the space of continuously differentiable functions up to the order k in Ω . Suppose (1.1) is a PDE of order m. By a solution of the equation (1.1) we mean a function u E C" ($2) such that the substitution of u and its derivatives up to the order m in (1.1) makes it an identity in (XI,.. ., xn) E Ω .
Various partial differential equations are given below:
Each one of these equations has two independent variables denoted either by z,y or z , t . Equations 1, 2 and 3 are of first-order. Equations numbered as 4, 5, 6, 8 , 9, 11 and 12 are of second-order; 10 is of third-order; 7 is of fourth-order. Examples 2, 3, 8, 9, 10, 11 and 12 are distinguished from the others in that they are not "linear".
Linearity means the following. The correspondence :
U(x1, x2, x3, ..xn) ---> Lu := F(x1, x2, x3.. xn, u, ux1,.. ux2, uxixj,..)
defines an operator L. The operator L is said to be linear iff (if and only if )
L (ClUl + c2u2) = ClLUl + c2Lu2 --(1.2)
for any functions u1, u2 and any constants c1, c2 E R.
Curl of a function in arbitrary coordinate systemThe curl of a function in Euclidean space can be generalized to any curved space as follows:
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