1. Calculus Review GLY-4822

2. Slope Slope = rise/run =  y/  x = (y 2 – y 1 )/(x 2 – x 1 ) Order of points 1 and 2 not critical, but keeping them together is Points may lie in any quadrant: slope will work out Leibniz notation for derivative based on  y/  x; the derivative is written dy/dx

3. Exponents x 0 = 1

4. Derivative of a line y = mx + b slope m and y axis intercept b derivative of y = ax n + b with respect to x: dy/dx = a n x (n-1) Because b is a constant -- think of it as bx 0 -- its derivative is 0bx -1 = 0 For a straight line, a = m and n = 1 so dy/dx = m 1 x (0), or because x 0 = 1, dy/dx = m

5. Derivative of a polynomial In differential Calculus, we consider the slopes of curves rather than straight lines For polynomial y = ax n + bx p + cx q + … derivative with respect to x is dy/dx = a n x (n-1) + b p x (p-1) + c q x (q-1) + …

6. Example y = ax n + bx p + cx q + … dy/dx = a n x (n-1) + b p x (p-1) + c q x (q-1) + …

7. Numerical Derivatives slope between points

8. Derivative of Sine and Cosine sin(0) = 0 period of both sine and cosine is 2  d(sin(x))/dx = cos(x) d(cos(x))/dx = -sin(x)

9. Partial Derivatives Functions of more than one variable Example: C(x,y) = x 4 + y 3 + xy

10. Partial Derivatives Partial derivative of h with respect to x at a y location y 0 Notation  h/  x| y=y0 Treat ys as constants If these constants stand alone, they drop out of the result If they are in multiplicative terms involving x, they are retained as constants

11. Partial Derivatives Example: C(x,y) = x 4 + y 3 + xy  C/  x| y=y 0 = 4x 3 + y 0

12. WHY?

13. Gradients del h (or grad h) Flow (Darcy’s Law):

14. Gradients del C (or grad C) Diffusion (Fick’s 1 st Law):

15. Basic MATLAB

16. Matlab Programming environment Post-processer Graphics Analytical solution comparisons Use File/Preferences/Font to adjust interface font size

17. Vectors >> a=[1 2 3 4] a = 1 2 3 4 >> a' ans = 1 2 3 4

18. Autofilling and addressing Vectors > a=[1:0.2:3]' a = 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000 2.2000 2.4000 2.6000 2.8000 3.0000 >> a(2:3) ans = 1.2000 1.4000

19. xy Plots >> x=[1 3 6 8 10]; >> y=[0 2 1 3 1]; >> plot(x,y)

20. Matrices >> b=[1 2 3 4;5 6 7 8] b = 1 2 3 4 5 6 7 8 >> b' ans = 1 5 2 6 3 7 4 8

21. Matrices >> b=2.2*ones(4,4) b = 2.2000 2.2000 2.2000 2.2000

22. Reshape >> a=[1:9] a = 1 2 3 4 5 6 7 8 9 >> bsquare=reshape(a,3,3) bsquare = 1 4 7 2 5 8 3 6 9 >>

23. Load a = load(‘filename’); (semicolon suppresses echo)

24. If if(1) … else … end

25. For for i = 1:10 … end

26. BMP Output bsq=rand(100,100); %bmp1 output e(:,:,1)=1-bsq; %r e(:,:,2)=1-bsq; %g e(:,:,3)=ones(100,100); %b imwrite(e, 'junk.bmp','bmp'); image(imread('junk.bmp')) axis('equal')

27. Quiver (vector plots) >> scale=10; >> d=rand(100,4); >> quiver(d(:,1),d(:,2),d(:,3),d(:,4),scale)

28. Contours h=[…]; Contour(h) Or Contour(x,y,h)

29. Contours w/labels C=[…]; [c,d]=contour(C); clabel(c,d), colorbar

30. Numerical Partial Derivatives slope between points MATLAB –h=[]; (order assumed to be low y on top to high y on bottom!) –[dhdx,dhdy]=gradient(h,spacing) –contour(x,y,h) –hold –quiver(x,y,-dhdx,-dhdy)

31. Gradient Function and Streamlines [dhdx,dhdy]=gradient(h); [Stream]= stream2(X,Y,U,V,STARTX,STARTY); [Stream]= stream2(-dhdx,- dhdy,[51:100],50*ones(50,1)); streamline(Stream) (This is for streamlines starting at y = 50 from x = 51 to 100 along the x axis. Different geometries will require different starting points.)

32. Stagnation Points

33. Integral Calculus

34. Integral Calculus: Special Case