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HARVARD MATH 21A - Homework

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3/3/2003, PARTIAL DERIVATIVES Math 21a, O. KnillHOMEWORK: Section 11.3: 48, 62, 66, 70, 76PARTIAL 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, or 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)GRADIENT. If f (x, y, z) is a function of three variables, then∇f(x, y, z) =∂∂xf(x, y, z),∂∂yf(x, y, z),∂∂zf(x, y, z)is called the gradient of f. The symbol ∇ is called Nabla.NORMAL. As we will see later, the gradient ∇f(x, y) is orthog-onal to the level curve f (x, y) = c and the gradient ∇f(x, y, z) isnormal to the level surface f(x, y, z). For example, the gradientof f(x, y, z) = x2+ y2− z2at a point (x, y, z) is (2x, 2y, −2z).WHERE ARE PARTIAL DERIVATIVES IMPORTANT?• Geometry. For example, the gradient ∇f(x, y, z) is a vector normal to a surface at the point (x, y, z).Tangent spaces.• Approximations, linarizations.• Partial differential equations. Laws which describe physics.• Optimization problems, as we will see later.• Solution to some integration problems using generalizations of fundamental theorem of calculus.• Generally helpful to understand and analyze functions of several variables.PARTIAL DIFFERENTIAL EQUATIONS. An equation which involves partial derivatives of an unknownfunction is called a partial differential equation. If only the derivative with respect to one variable appears,it is called an ordinary differential equation.1) fxx(x, y) = fyy(x, y) is an example of a partial differential equation 1) fx(x, y) = fxx(x, y) would be anordinary differential equation (the variable y can be considered as a parameter.LAPLACE.fxx+ fyy= 0 .f(x, t) = x2− y2.ADVECTION.ft= fx.f(x, t) = g(x − t).WAVES.ftt= fxx. f(t, x) =sin(x − t) + sin(x + t)HEATft= fxxf(t, x) =1√te−x2/(4t)”A great deal of my work is just playing with equations and seeing what they give. I don’tsuppose that applies so much to other physicists; I think it’s a peculiarity of myself that Ilike to play about with equations, just looking for beautiful mathematical relationswhich maybe don’t have any physical meaning at all. Sometimes they do.” - Paul A. M.Dirac.Dirac discovered a PDE describing the electron which is consistent both with quantum theory and special relativity.This won him the Nobel Prize in 1933. Dirac’s equation could have two solutions, one for an electron with positiveenergy, and one for an electron with negative energy. Dirac interpretated the later as an antiparticle: the existence ofantiparticles was later


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HARVARD MATH 21A - Homework

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