Let w = e®(z+=) dy ^ dz+e" dz ^ dæ + e(z =) dæ A dy be a two-form on R³, and o : (IR>0)² → R³ be the smooth function $(u, v) = (In(uv), In(u+ v), In(uv)). Find o' w. d*w=| |du ^ dv
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- Bounded by the cylinder x 2 + y 2 = 1 and the planes y= z x =0 z= 0 in the first octantLet ƒ(x, y) =(x2-y2)/(x2+y2) for(x, y) ≠ (0, 0). Is it possible to define ƒ(0, 0) in a way that makes ƒ continuous at the origin? WhyLet f (x, y) = 2x^3 - 6xy + 3y^2 be a function defined on xy-plane (a) Find first and second partial derivatives of.(b) Determine the local extreme points of f (max., min., saddle points) if there are any.(c) Find the absolute max. and absolute min. values of f over the closed region bounded by the linesx = 2, y = 0, and y = x
- Let, z=sinx cosy, where x=s+t, y=s-t,andz=x^2+y^2, x=sin(st), y=s^2t^3,Determine the partial derivatives; ??/?? ??? ??/??.A function f is called homogeneous of degree n if it satisfies the equation f(tx, ty) = tnf(x, y) for all t, where n is a positive integer and f has continuous second-order partial derivatives. If f is homogeneous of degree n, show that fx(tx, ty) = tn − 1fx(x, y). (Hint: Use Chain Rule)Suppose that a thin metal plate of area A and constant density doccupies a region R in the xy-plane, and let My be the plate’s momentabout the y-axis. Show that the plate’s moment about the linex = b is My - bδA if the plate lies to the right of the line, and
- A bug crawls on the surface z = x2 - y2 directly above a path in the xy-plane given by x = ƒ(t) and y = g(t). If ƒ(2) = 4, ƒ′(2) = -1, g(2) = -2, and g′(2) = -3, then at what rate is the bug’s elevation z changing when t = 2?Suppose that the second order partial derivatives, fx,y and fy,x, are both continuous on an open set V in R2. Use Fubini’s theorem to prove that fx,y = fy,x in V . Hint: if fx,y(a) − fy,x(a) > 0, there is a rectangle R containing a on which fx,y − fy,x > 0.Let X, Y ∼ U(0, 1) and X and Y independent. Find the PDF of U = min(X, Y ).
- Let F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that encloses a region of area 16 (Figure 20). Calculate ∮C F • dr, where C is oriented in the counterclockwise direction (when viewed from above the plane).A function f has continuous second partial derivatives on an open region containing the critical point (a, b). If fxx(a, b) and fyy(a, b) have opposite signs, what is implied? Explain.Show that the function f:R^(2)->R defined by f(x,y)={((xy^(2))/(x^(2)+y^(4)), if x^(2)+y^(4)!=0),(0, if x=0=y):} possess first order partial derivatives everywhere including the origin but the function is discontinuous at the origin