Lecture 11: 3/5/2004, TANGENT PLANES Math21aHOMEWORK. 11.5: 34,38,58,60REMINDER: TANGENT LINE. Because ~n = ∇f(x0, y0) = ha, bi is perpendicular to the level curve f(x, y) = cthrough (x0, y0), the equation for the tangent line isax + by = d, a = fx(x0, y0), b = fy(x0, y0), d = ax0+ by0Example: Find the tangent to the graph of the function g(x) = x2at the point (2, 4). Solution: the level curve f(x, y) = y − x2= 0is the graph of a function g(x) = x2and the tangent at a point(2, g(2)) = (2, 4) is obtained by computing the gradient ha, bi =∇f(2, 4) = h−g0(2), 1i = h−4, 1i and forming −4x + y = d, whered = −4·2+1·4 = −4. The answer is−4x + y = −4 which is theline y = 4x − 4 of slope 4. Graphs of 1D functions are curves inthe plane, you have computed tangents in single variable calculus.-3 -2 -1 1 2 3-6-4-22468GRADIENT IN 3D. If f(x, y, z) is a function of three variables, then ∇f(x, y, z) =(fx(x, y, z), fy(x, y, z), fz(x, y, z)) is called the gradient of f.POTENTIAL AND FORCE. Force fields F in nature often are gradients of a function U(x, y, z). The functionU is called a potential of F or the potential energy.EXAMPLE. If U (x, y, z) = 1/|x|, then ∇U(x, y, z) = −x/|x|3. The function U(x, y, z) is the Coulomb poten-tial and ∇U is the Coulomb force. The gravitational force has the same structure but a different constant.While much weaker, it is more effective because it only appears as an attractive force, while electric forces canbe both attractive and repelling.LEVEL SURFACES. If f(x, y, z) is a func-tion of three variables, then f(x, y, z) = Cis a surface called a level surface of f . Thepicture to the right shows the Barth surface(3 + 5t)(−1 + x2+ y2+ z2)2(−2 + t + x2+y2+ z2)28(x2−t4y2)(−t4x2+ z2) (y2−t4z2)(x4− 2x2y2+ y4− 2x2z2− 2y2z2+ z4) = 0,where t = (√5 − 1)/2 is the golden ratio.ORTHOGONALITY OF GRADIENT. We have seenthat the gradient ∇f (x, y) is normal to the level curvef(x, y) = c. This is also true in 3 dimensions:The gradient ∇f (x, y, z) is nor-mal to the level surface f(x, y, z).The argument is the same as in 2 dimensions: take a curve~r(t) on the level surface. Thenddtf(~r(t)) = 0. The chainrule tells from this that ∇f(x, y, z) is perpendicular to thevelocity vector ~r0(t). Having ∇f tangent to all tangentvelocity vectors on the surfaces forces it to be orthogonal.EXAMPLE. The gradient of f(x, y, z) = x2+ 2y2+ z2ata point (x, y, z) is (2x, 4y, 2z). It illustrates well that goinginto the direction of the gradient increases the value of thefunction.TANGENT PLANE. Because ~n = ∇f (x0, y0, z0) = ha, b, ciis perpendicular to the level surface f(x, y, z) = C through(x0, y0, z0), the equation for the tangent plane isax + by + cz = d, (a, b, c) =∇f(x0, y0, z0), d = ax0+ by0+ cz0.EXAMPLE. Find the general formula for the tangent planeat a point (x, y, z) of the Barth surface. Just kidding ...Note however that computing this would be no big deal withthe help of a computer algebra system like Mathematica.Lets look instead at the quartic surfacef(x, y, z) = x4− x3+ y2+ z2= 0which is also called the ”piriform” or ”pair shaped surface”.What is the equation for the tangent plane at the point P =(2, 2, 2)? We get ha, b, ci = h20, 4, 4i and so the equation ofthe plane 20x + 4y + 4z = 56.EXAMPLE. An important example of a level surface isg(x, y, z) = z − f(x, y) which is the graph of a function oftwo variables. The gradient of g is ∇f = (−fx, −fy, 1).This allows us to find the equation of the tangent plane ata point.Quizz: What is the relation between the gradient of f inthe plane and the gradient of g in
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