1. [FX,FY,FZ,...] = GRADIENT(F,...) [FX,FY,FZ,…] is numerical gradient of matrix F Distance between grid points in X,Y,Z,… direction FX corresponds to dF/dx, the differences in the x (column) direction

2. [x,y] = meshgrid(-2:.2:2, -2:.2:2); z = x.* exp(-x.^2 - y.^2); [px,py] = gradient(z,.2,.2); contour(z),hold on, quiver(px,py), hold off [x,y] = meshgrid(-2:1:2, -2:1:2) Create a mesh on the domain of the function z. Examples: x(4,5)=1 y(4,5)=2 x(2,2)=-1 y(2,5)=2 (1,1) (1,2) (1,3) (1,4) (1,5) (2,1) (2,2) (2,3) (2,4) (2,5) (3,1) (3,2) (3,3) (3,4) (3,5) (4,1) (4,2) (4,3) (4,4) (4,5) (5,1) (5,2) (5,3) (5,4) (5,5) x y

3. (1,1) (1,2) (1,3) (1,4) (1,5) (2,1) (2,2) (2,3) (2,4) (2,5) (3,1) (3,2) (3,3) (3,4) (3,5) (4,1) (4,2) (4,3) (4,4) (4,5) (5,1) (5,2) (5,3) (5,4) (5,5) x y [x,y] = meshgrid(-2:.2:2, -2:.2:2); z = x.* exp(-x.^2 - y.^2); [px,py] = gradient(z,.2,.2); contour(z),hold on, quiver(px,py), hold off z=x.*exp(-x.^2-y.^2) Element by element operation X(4,5)=1 Y(4,5)=2 Z(4,5)=1*exp(-1^2-2^2)=exp(-5)=0.0067 -0.0007 -0.0067 0 0.0067 0.0007 -0.0135 -0.1353 0 0.1353 0.0135 -0.0366 -0.3679 0 0.3679 0.0366 -0.0135 -0.1353 0 0.1353 0.0135 -0.0007 -0.0067 0 0.0067 0.0007

4. (1,1) (1,2) (1,3) (1,4) (1,5) (2,1) (2,2) (2,3) (2,4) (2,5) (3,1) (3,2) (3,3) (3,4) (3,5) (4,1) (4,2) (4,3) (4,4) (4,5) (5,1) (5,2) (5,3) (5,4) (5,5) x y [x,y] = meshgrid(-2:.2:2, -2:.2:2); z = x.* exp(-x.^2 - y.^2); [px,py] = gradient(z,.2,.2); contour(z),hold on, quiver(px,py), hold off -0.0007 -0.0067 0 0.0067 0.0007 -0.0135 -0.1353 0 0.1353 0.0135 -0.0366 -0.3679 0 0.3679 0.0366 -0.0135 -0.1353 0 0.1353 0.0135 -0.0007 -0.0067 0 0.0067 0.0007 [px,py]=gradient(z,1,1) px = -0.0061 0.0003 0.0067 0.0003 -0.0061 -0.1219 0.0067 0.1353 0.0067 -0.1219 -0.3312 0.0183 0.3679 0.0183 -0.3312 -0.1219 0.0067 0.1353 0.0067 -0.1219 -0.0061 0.0003 0.0067 0.0003 -0.0061 py = -0.0128 -0.1286 0 0.1286 0.0128 -0.0180 -0.1806 0 0.1806 0.0180 0 0 0 0 0 0.0180 0.1806 0 -0.1806 -0.0180 0.0128 0.1286 0 -0.1286 -0.0128

5. CONTOUR(Z) is a contour plot of matrix Z treating the values in Z as heights above a plane. A contour plot are the level curves of Z for some values V. The values V are chosen automatically. px = -0.0061 0.0003 0.0067 0.0003 -0.0061 -0.1219 0.0067 0.1353 0.0067 -0.1219 -0.3312 0.0183 0.3679 0.0183 -0.3312 -0.1219 0.0067 0.1353 0.0067 -0.1219 -0.0061 0.0003 0.0067 0.0003 -0.0061 py = -0.0128 -0.1286 0 0.1286 0.0128 -0.0180 -0.1806 0 0.1806 0.0180 0 0 0 0 0 0.0180 0.1806 0 -0.1806 -0.0180 0.0128 0.1286 0 -0.1286 -0.0128 z = x.* exp(-x.^2 - y.^2); [x,y] = meshgrid(-2:.2:2, -2:.2:2); z = x.* exp(-x.^2 - y.^2); [px,py] = gradient(z,.2,.2); contour(z),hold on, quiver(px,py), hold off

6. [x,y] = meshgrid(-2:.2:2, -2:.2:2); z = x.* exp(-x.^2 - y.^2); [px,py] = gradient(z,.2,.2); contour(z),hold on, quiver(px,py), hold off HOLD ON holds the current plot and all axis properties so that subsequent graphing commands add to the existing graph. HOLD OFF returns to the default mode whereby PLOT commands erase the previous plots and reset all axis properties before drawing new plots.

7. DIV = DIVERGENCE(X,Y,Z,U,V,W) computes the divergence of a 3-D vector field U,V,W. The arrays X,Y,Z define the coordinates for U,V,W and must be monotonic and 3-D plaid (as if produced by MESHGRID). load wind div = divergence(x,y,z,u,v,w); slice(x,y,z,div,[90 134],[59],[0]); shading interp daspect([1 1 1]) camlight

8. load wind div = divergence(x,y,z,u,v,w); slice(x,y,z,div,[90 134],[59],[0]); shading interp daspect([1 1 1]) camlight Wind is an array of (41,35,15). x z y Slice at (90,59,0) Origin: (70.2,17.5,0) End: (134.3, 60, 16)

9. load wind div = divergence(x,y,z,u,v,w); slice(x,y,z,div,[90 134],[59],[0]); shading interp daspect([1 1 1]) camlight

10. load wind div = divergence(x,y,z,u,v,w); slice(x,y,z,div,[90 134],[59],[0]); shading interp daspect([1 1 1]) camlight

11. load wind div = divergence(x,y,z,u,v,w); slice(x,y,z,div,[90 134],[59],[0]); shading interp daspect([1 1 1]) camlight

12. [CURLX, CURLY, CURLZ, CAV] = CURL(X,Y,Z,U,V,W) computes the curl and angular velocity perpendicular to the flow (in radians per time unit) of a 3D vector field U,V,W. The arrays X,Y,Z define the coordinates for U,V,W and must be monotonic and 3D plaid (as if produced by MESHGRID). Ω

13. load wind cav = curl(x,y,z,u,v,w); slice(x,y,z,cav,[90 134],[59],[0]); shading interp daspect([1 1 1]); axis tight colormap hot(16) camlight

14. L = DEL2(U), when U is a matrix, is a discrete approximation of 0.25*del^2 u = (d^2u/dx^2 + d^2/dy^2)/4. The matrix L is the same size as U, with each element equal to the difference between an element of U and the average of its four neighbors.