Question

Find the second order partial derivative
to x of F(x,y) = cosx + xyety + xsiny.​

Answer 1

Step-by-step explanation:

Let F=[A , B , C] . Then Fx=[Ax , Bx , Cx] , Fxx=[Axx , Bxx , Cxx] , Fy=[Ay , By , Cy] ,

Fyy=[Ayy , Byy , Cyy] , Fxy=[Axy , Bxy , Cxy] , Fyx=[Ayx , Byx , Cyx]. Note that F is a function from R^2 to R^3 ,then the(total) derivative of F is a 3.2 matrix and the second derivative of F is a 3.2.2 ( 3row , 2column,2 height) space matrix which is not a natural object. So there is a notion of tensor as a tool to describe derivative of orders 2, 3 ,…n. Therefore we can understand the differential is a better tool than the derivative. For example if W=G(x , y) is a two variables real function, we have