[fx,fy,fz,...] = gradient(f,...) [fx,fy,fz,…] is numerical gradient of matrix f distance between...

[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

[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)=1y(4,5)=2x(2,2)=-1y(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

(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

212

105

23

41*

53

12

109

42

23

41*.

53

12

X(4,5)=1Y(4,5)=2Z(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

(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 -0.0061py = -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

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.

1 1.5 2 2.5 3 3.5 4 4.5 51

1.5

2

2.5

3

3.5

4

4.5

5

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 -0.0061py = -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

1 1.5 2 2.5 3 3.5 4 4.5 51

1.5

2

2.5

3

3.5

4

4.5

5

[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.

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 winddiv = divergence(x,y,z,u,v,w);slice(x,y,z,div,[90 134],[59],[0]); shading interpdaspect([1 1 1])camlight

load winddiv = divergence(x,y,z,u,v,w);slice(x,y,z,div,[90 134],[59],[0]); shading interpdaspect([1 1 1])camlight

Wind is an array of (41,35,15).

x

z

y

Slice at (90,59,0)

6080

100120

140

0

20

40

60-5

0

5

10

15

20Origin: (70.2,17.5,0)End: (134.3, 60, 16)

load winddiv = divergence(x,y,z,u,v,w);slice(x,y,z,div,[90 134],[59],[0]); shading interpdaspect([1 1 1])camlight

6080

100120

140

0

20

40

60-5

0

5

10

15

20

6080

100120

140

0

20

40

60-5

0

5

10

15

20

6080

100120

140

0

20

40

60-5

0

5

10

15

20

load winddiv = divergence(x,y,z,u,v,w);slice(x,y,z,div,[90 134],[59],[0]); shading interpdaspect([1 1 1])camlight

60

80

100

120

140

0

20

40

60-20

-10

0

10

20

load winddiv = divergence(x,y,z,u,v,w);slice(x,y,z,div,[90 134],[59],[0]); shading interpdaspect([1 1 1])camlight

60

80

100

120

140

0

20

40

60-20

-10

0

10

20

-6

-4

-2

0

2

4

6

[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).

v

Rv

2

1

Ω

load windcav = curl(x,y,z,u,v,w);slice(x,y,z,cav,[90 134],[59],[0]); shading interpdaspect([1 1 1]); axis tightcolormap hot(16)camlight

8090

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120130

20

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50

0

5

10

15

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.