10/22/2002, PARTIAL DERIVATIVES (chapter 11) Math 21a, O. KnillHomework for Thursday: 11.1 24, 31-36 11.2 14,28,34LEVEL CURVES2D: If f(x, y) is a function of two variables, thenf(x, y) = const is curve in the plane. It iscalled contour line or level curve. For example,f(x, y) = 4x2+ 3y2= 1 is an ellipse. Level curvesallow to visualize the function f.LEVEL SURFACES.3D: If f(x, y, z) is a function of three variables and cis a constant then f(x, y, z) = c is a surface in space.It is called a contour surface or a level survae.For example if f(x, y, z) = 4x2+ 3y2+ z2, then thecontour surfaces are ellipsoids.EXAMPLE. Let f(x, y) = x2− y2. The set x2− y2= 0 is the union of the sets x = y and x = −y. The setx2− y2= 1 consists of two hyperbola with with their tips at (−1, 0) and (1, 0). The set x2− y2= −1 consistsof two hyperbola with their tips at (0, ±1).EXAMPLE. Let f(x, y, z) = x2+ y2− z2. f(x, y, z) =0, f(x, y, z) = 1, f(x, y, z) = −1. The set x2+y2−z2= 0 is a conerotational symmetric around the z-axes. The set x2+ y2− z2= 1is a one-sheeted hyperboloid, the set x2+ y2− z2= −1 is atwo-sheeted hyperboloid. (How to see that it is two-sheeted:the intersection with z = c is empty for −1 ≤ z ≤ 1.)CONTOUR MAP. Drawing several contour lines or surfaces produces a contour map.TOPOGRAPHY. Topographical maps often show the curves of equal height. With this information, it is usuallyalready possible to have a good picture of the situation.SPECIAL LINESIsobars: pressureIsoclines: directionIsothermes: temperatureIsoheight: heightFor example, the isobars to the right show the lines of constant pressuretoday in Europe.PARTIAL DERIVATIVE. If f (x, y, z) is a function of three variables, then∂∂xf(x, y, z) is defined as thederivative of the function g(x) = f(x, y, z), where y and z are fixed. The other derivatives with respect to yand z are defined similarly.REMARK. The partial derivatives measure the rate of change of the function in the x, y, z directions.NOTATION. One also writes fx(x, y, z) =∂∂xf(x, y, z) etc. For iterated derivatives the notation is similar: forexample fxy=∂∂x∂∂yf.EXAMPLE. f (x, y) = x4− 6x2y2+ y4. We have fx(x, y) = 4x3−12xy2, fxx= 12x2−12y2, fy(x, y) = −12x2y +4y3, fyy= −12x2+12y2.We see that fxx+ fyy= 0. A function which satisfies this equation iscalled harmonic. The equation itself is called a partial differentialequation (see separate handout).CLAIROT THEOREM. If fxyand fyxare both continuous, then fxy= fyx. Proof. Compare the two sides:dxfx(x, y) ∼ f(x + dx, y) − f(x, y)dydxfxy(x, y) ∼ f(x+ dx, y + dy) − f(x + dx, y + dy)−(f(x + dx, y) − f(x, y))dyfy(x, y) ∼ f(x, y + dy) − f (x, y).dxdyfxy(x, y) ∼ f (x + dx, y + dy) − f(x + dx, y) −(f(x, y + dy) − f (x, y))CONTINUITY IS NECESSARY. Example: f(x, y) = (x3y − xy3)/(x2+ y2) contradicts Clairot:fx(x, y) = (3x2y−y3)/(x2+y2)−2x(x3y−xy3)/(x2+y2)2, fx(0, y) = −y, fxy(0, 0) = −1,fy(x, y) = (x3−3xy2)/(x2+y2)−2y(x3y−xy3)/(x2+y2)2, fy(x, 0) = x2, fy,x(0, 0) = 1.f(x, y) fx(x, y)fy(x, y) fxy(x, y)ABOUT CONTINUITY. In reality, one sometimes has to deal with functions which are not smooth: Forexample, when plotting the temperature of water in relation to pressure and volume, one experiences phasetransitions, an other example are water waves breaking in the ocean. Mathematicians have also tried toexplain ”catastrophic” events mathematically with a theory called ”catastrophe theory”. Discontinuous thingsare useful (for example in switches), or not so useful (for example, if something
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