Calculus Review GLY-5826

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

Exponents x 0 = 1

Derivative of ax n y = ax n derivative of y = ax n with respect to x: –dy/dx = a n x (n-1)

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

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) + …

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) + …

Numerical Derivatives Slope between points Examples

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)

Higher Order Derivatives Second derivative: –d 2 y/dx 2 = d(dy/dx)/dx –Note positions of the ‘twos’; dimensionally consistent Practical: –Take derivative –Take derivative again –d 2 (x 3 )/dx 2 = d(3x 2 )/dx = 6x

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

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

Partial Derivatives Example: –h(x,y) = x 4 + y 3 + xy – ∂ h/ ∂ x = 4x 3 + y – ∂ h/ ∂ x| y=y 0 = 4x 3 + y 0

Partial Derivatives Example: –h(x,y) = x 4 + y 3 + xy – ∂ h/ ∂ y = 3y 2 + x – ∂ h/ ∂ y| x=x 0 = 3y 2 + x 0

WHY?

Gradients del C (or grad C) Darcy’s Law:

Basic MATLAB

Matlab Programming environment Post-processer Graphics Analytical solution comparisons

Vectors >> a=[ ] a = >> a' ans =

Autofilling and addressing Vectors > a=[1:0.2:3]' a = >> a(2:3) ans =

xy Plots >> x=[ ]; >> y=[ ]; >> plot(x,y)

Matrices >> b=[ ; ] b = >> b' ans =

Matrices >> b=2.2*ones(4,4) b =

Reshape >> a=[1:9] a = >> bsquare=reshape(a,3,3) bsquare = >>

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

If if(1) … else … end

For for i = 1:10 … end

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')

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

Contours h=[…]; Contour(h)

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

Numerical Partial Derivatives slope between points MATLAB –h=[]; –[dhdx,dhdy]=gradient(h) –contour([1:20],[1:20],h) –hold –quiver([1:20],[1:20],-dhdx,-dhdy)

Gradient Function and Streamlines [dhdx,dhdy]=gradient(h); [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.)

Stagnation Points

Integral Calculus

Integral Calculus: Special Case