MATA35H3 Lecture 21: Lecture 21

Find the (possibly complex) Eigen values of the matrix Find the corresponding eigenspaces (which are subspaces of C3). By finding a basis B for C3 consisting of eigenvectors of f : x rightarrow Ax, find a diagonal matrix D with [f]B = D Find a (complex) matrix P such that A = PDP-1, where D is the diagonal matrix in (c), and check your answer by checking that AP = PD. Hint. Let P = PB where D is the basis found in (c). det (A lambda I3)= det So the Eigen values, which are the solutions of det(A - lambda I3) = 0, are 2,-3i,3i. The eigenspace corresponding to 2 consists of those vectors (r, y, z) epsilon C3 such that i.e. -2x + 3z = 0,0 = 0, -3x - 2z = 0. From a Gaussian elimination or simple manipulation x = z = 0. So the eigenspace is {(0, y,0):y epsilon C}. The eigenspace corresponding to 3i consists of those vectors (r, y. z) epsilon C3 such that i.e. -3ix + 3z = 0, (2 - 3i)y = 0, -3x - 3iz = 0. i.e. x = -iz.y = 0. x = - iz. So the eigenspace is {(-iz,0. z) : z epsilon C} Replacing i by -i in the calculation above, the eigenspace corresponding to -3i is {(iz,0,2): z epsilon C}. Let B = {(0,1,0), (-i,0,l),(i,0,l)}. Then f(0,1,0) = (2,0,0) = 2(1,0,0),f(-i,0,1) = (3,0,3i) = 3t(-i,0,l),f(i,0,l) (3,0, -3i) = -3i(t',0,1).

Justify your work, communicate clearly and mathematically. calculation not allowed. Unless decimal approximations are requested, numerical answers must Label your problems clearly and mark the work you want graded/not graded; also, box in your answers. Using techniques from this class, approximate the value of exy - cosx at (.1,2.3) Consider the vector field F(x, y) = (cos(sin x + y) cosx ex, cos(sin x + y) + y) the work done as you traverse the Archimedes spiral (r = Theta) from (x,y) = (0,0) to (x,y)= (2 pi. 0). (Hint: check to see if the vector field is conservative.) Find the absolute maximum and minimum of y ex-z on the ellipsoid9x2+4y2+36z2=36 Compute Consider the region bounded by 2 = 4 - x2 - y2 and 2 = 0. Let S be the boundary of this region. Compute the flux of F = (xy2z - xz, yz + ev, - zev) across S. If F represents a fluid flow, explain what the above computation means. First, sketch and describe the surface 5: r(u. v) - (ucosv, usinv.v,v), where u Second, let f(x. y. z) be the distance function from the point (x. y, z) to the z - axi% Integrate the function f over S. Pick 2 of the following and solve. Compute fc(:2 + yeI)dx+ (ez + yz)dy + (2xz + y2/2 )dz along the curve C: r(t)= 5 cost, sint, cost, sint). > Consider the curve coordinates this is just the wobbly circle r = 3 + cos St). Compute where F - (-y,x)/(x2 + y2). State and prove Clairaut's theorem. State and prove the chain rule. State and prow Green s theorem for simply-connected regions. Bonus if you can prove Green's theorem for non-simply-connected regions

(1 point) Suppose F(z,y)-3sin(5) sin(5):-3cos(5)cos and C is the curve from P to Q in the figure. Calculate the line integral of F along the curve C. The labeled points are P = (-2,2),0-(-2,一2).R-(, and SQ are trigonometric functions of period 2π and amplitude 1 , and S = (T,-T),The curves PR -5 3-21 Click on graph to enlarge)