6 Points Let F be the vector field represented in the figure: P(-1,1) 1907/1X X Q3.1...
Let F be the vector field represented in the figure: y X 1Q0Y, 1X P(-1, 1) X Q3.1 3 Points 2d-Curl F(0,0) > 0 O 2d-Curl F(0,0) = 0 2d-Curl F(0,0) < 0 Q3.2 3 Points OV: F(0,0) > 0 OV: F(0,0) = 0 OV: F(0,0) < 0
Q3 6 Points Let F be the vector field represented in the figure: P(-1,1) toyIX Q3.1 3 Points O2d-Curl F(0,0) > 0 O2d-Curl F(0,0) = 0 O2d-Curl F(0,0) <0 Q3.2 3 Points OV. F(0,0) > 0 OV. F(0,0) = 0 OV. F(0,0) < 0
Consider the vector field F (x, y, z) = <y?, z2, x?>. Compute the curl (F). Use Stokes' Theorem to evaluate S. F. dr where C is the triangle (0,0,0), (1,0,0), and (0, 1, 1) oriented counter-clockwise when viewed from above.
(d). Let X, X,...,x be a random sample from the Normal(0,0) distribution, 0 >0. Find the uniformly most powerful test for H:050 versus H,:0>
Let F(XYZ) = <2y27, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative, find a function f such that F = vfand f(1,2,1) = 8. (b) Using the result of part(a), evaluate the line integral of F along the following curve C from (0,0,0) to (3.9, 1.8, 2.3). y2 + x4z2 + 2x4(x3 + y2 + 24)1/2 = K Kis a constant .- Answer:
Let F be a field of characteristic p > 0. Show that f = t4 +1 € F[t] is not irreducible. Let K be a splitting field of f over F. Determine which finite field F must contain so that K = F.
Let T: P2 --> R2 be the linear transformation such that T(x+1)=(1,1), T(x2)=(1,0) and T(x-1)=(0, 1). Find T(2+x+x2).
Given z = 2 y2 – 3xy , find the slope of the surface at (1,1,-1) in the direction of ū =< 2,3>
96. Consider a vector field F(x, y, z) =< x + x cos(yz), 2y - eyz, z- xy > and scalar function f(x, y, z) = xy3e2z. Find the following, or explain why it is impossible: a) gradF (also denoted VF) b) divF (also denoted .F) c) curl(f) (also denoted xf) d) curl(gradf) (also denoted V x (0f) e) div(curlF) (also denoted 7. (V x F))
Let f : [0,1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2 - y<== f() - f(y)< € for every 2, Y € [0,1]. The graph of f is the set Gj = {, f(c)): 1 € [0,1]}. : Show that G has measure zero.