Calculus Review

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Calculus Review
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Slope

• Slope rise/run
• Dy/Dx
• (y2 y1)/(x2 x1)
• 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 Dy/Dx
  the derivative is written dy/dx

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Exponents

• x0 1

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Derivative of a line

• y mx b
• slope m and y axis intercept b
• derivative of y axn b with respect to x
• dy/dx a n x(n-1)
• Because b is a constant -- think of it as bx0 --
  its derivative is 0bx-1 0
• For a straight line, a m and n 1 so
• dy/dx m 1 x(0), or because x0 1,
• dy/dx m

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Derivative of a polynomial

• In differential Calculus, we consider the slopes
  of curves rather than straight lines
• For polynomial y axn bxp cxq
• derivative with respect to x is
• dy/dx a n x(n-1) b p x(p-1) c q x(q-1)

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Example
y axn bxp cxq
dy/dx a n x(n-1) b p x(p-1) c q x(q-1)

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Numerical Derivatives

• slope between points

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Derivative of Sine and Cosine

• sin(0) 0
• period of both sine and cosine is 2p
• d(sin(x))/dx cos(x)
• d(cos(x))/dx -sin(x)

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Partial Derivatives

• Functions of more than one variable
• Example C(x,y) x4 y3 xy

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Partial Derivatives

• Partial derivative of h with respect to x at a y
  location y0
• Notation ?h/?xyy0
• 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

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Partial Derivatives

• Example
• C(x,y) x4 y3 xy
• ?C/? xyy0 4x3 y0

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WHY?
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Gradients

• del h (or grad h)
• Flow (Darcys Law)

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Gradients

• del C (or grad C)
• Diffusion (Ficks 1st Law)

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Basic MATLAB/Python
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Matlab/Python

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

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Vectors/Lists and tuples

• gtgt a1 2 3 4
• a
• 1 2 3 4
• gtgt a'
• ans
• 1
• 2
• 3
• 4

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Autofilling and addressing Vectors

• gt a10.23'
• a
• 1.0000
• 1.2000
• 1.4000
• 1.6000
• 1.8000
• 2.0000
• 2.2000
• 2.4000
• 2.6000
• 2.8000
• 3.0000
• gtgt a(23)
• ans

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xy Plots

• gtgt x1 3 6 8 10
• gtgt y0 2 1 3 1
• gtgt plot(x,y)

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Matrices

• gtgt b1 2 3 45 6 7 8
• b
• 1 2 3 4
• 5 6 7 8
• gtgt b'
• ans
• 1 5
• 2 6
• 3 7
• 4 8

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Matrices

• gtgt b2.2ones(4,4)
• b
• 2.2000 2.2000 2.2000 2.2000
• 2.2000 2.2000 2.2000 2.2000
• 2.2000 2.2000 2.2000 2.2000
• 2.2000 2.2000 2.2000 2.2000

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Reshape

• gtgt a19
• a
• 1 2 3 4 5 6 7 8
  9
• gtgt bsquarereshape(a,3,3)
• bsquare
• 1 4 7
• 2 5 8
• 3 6 9
• gtgt

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Load

• a load(filename) (semicolon suppresses echo)

anp.loadtxt('book3.csv',delimiter',')
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If

• if(1)
• else
• end

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For

• for i 110
• end

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BMP Output

• bsqrand(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')

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Quiver (vector plots)

• gtgt scale10
• gtgt drand(100,4)
• gtgt quiver(d(,1),d(,2),d(,3),d(,4),scale)

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Contours

• h
• Contour(h)
• Or Contour(x,y,h)

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Contours w/labels

• C
• c,dcontour(C)
• clabel(c,d), colorbar

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Numerical Partial Derivatives

• slope between points
• MATLAB
• h (order assumed to be low y on top to high y
  on bottom!)
• dhdx,dhdygradient(h,spacing)
• contour(x,y,h)
• hold
• quiver(x,y,-dhdx,-dhdy)

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Gradient Function and Streamlines

• dhdx,dhdygradient(h)
• Stream stream2(X,Y,U,V,STARTX,STARTY)
• Stream stream2(-dhdx,-dhdy,51100,50ones(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.)

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Stagnation Points
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Integral Calculus
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Integral Calculus Special Case
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Integral Calculus Special Case