Calculus Review

Slope Slope = rise/run = Dy/Dx = (y2 – y1)/(x2 – x1) Order of points 1 and 2 not critical Points may lie in any quadrant: slope will work out Leibniz notation for derivative based on Dy/Dx; the derivative is written dy/dx

Exponents x0 = 1

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 0b-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

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) + …

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

Numerical Derivatives slope between points

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

Partial Derivatives Functions of more than one variable Example: h(x,y) = x4 + y3 + xy

Partial Derivatives Partial derivative of h with respect to x at a y location y0 Notation dh/dx|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) = x4 + y3 + xy dh/dx|y=y0 = 4x3 + y0

WHY?

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

Numerical Derivatives slope between points MATLAB c=[]; [dcdx,dcdy]=gradient(c) contour([1:20],[1:20],c) hold quiver([1:20],[1:20],-dcdx,-dcdy)

Mathematica

Mathematica