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- (b)Find the derivative of g in the direction of ā = 2i+ curl w at point (0,-1,1): where 8 = In- u + 2 = z(xy+z? } , w = cos -i+ xe*lj+ xyz k U = z W = COS- yzarrow_forwardFind the flow of the velocity field F =(x + y)i - (x2 + y2)j along each of the following paths from(1, 0) to (-1, 0) in the xy-plane. The line segment from (1, 0) to (-1, 0)arrow_forwardConsider the function f(x,y) = (e* - x) cos(y). Suppose S is the surface z = f(x, y). (a) Find a vector which is perpendicular to the level curve of f through the point (1,4) in the direction in which f decreases most rapidly. vector =. (b) Suppose v = 67+3}+ak is a vector in 3-space which is tangent to the surface S at the point P lying on the surface above (1,4). What is a?arrow_forward
- Evaluate curl of A and divergence of curl of A, if A = x²yax + y²zay - 2xzaz Evaluate divergence VA when A= 2xay+3yay- 4zaz and V = xyz.arrow_forwardFind all the integral curves of the vector fields, indicate the domains of each vector field, and obtain two integral surfaces in each case (a) V = (y,-3,0), (b) V = (1, y, ry(22 + 1)). %3D %3Darrow_forwardQ3: ( (a) Prove that div(ū x w)= w•curl ū- ū• curl w functions. where u and w are vector (b)Find the derivative of g in the direction of a 2i+curl w at point (0,-1,1), where g = In- u +2 u = z(xy + z?, W = cos-i+xe"*l yzarrow_forward
- Let F and G be vector-valued functions such that - ₹(t) = (cos(ït), e²t−1, t² − 1), Ġ(1) = (1,1,−1), Ġ'(1) = (2,3,2), Ġ″(1) = (0,1,0) Find a vector equation of the tangent line to the graph of ₹ at (−1, e, 0).arrow_forwardA line of flux of a vector field F is a vector curve r (t) that satisfies equality: dr/dt=F(r(t)) If F represents the velocity field of a particle, then the lines of flux they correspond to the paths made by the particle. If r(t) =(e2t, ln|t|,1/t), t>0 then verify that r(t) is a line of flux of the vector field F(x, y, z) = (2x, z, −z2)arrow_forwardConsider the vectorial V=2+ŷ+ 2 z . . function=z²+x²y + y²2 and the gradient operator Please explicitly evaluate Vxarrow_forward
- Show that there is no vector field G = (P,Q, R) where P, Q and R have continuous second order partial derivatives such that curl G = (r", y, 2:"). %3Darrow_forward(a) Let C be the curve of intersection of the surfaces x^2 = 2y and 3z = xy. Find the length of C from the origin to the point (6, 18, 36). 4. (b) Let f be a continuous vector field which is parallel to the unit tangent vector at each point of a smooth curve C'. Show that f. dr L || f| ds . (c) Let C" be a simple closed piecewise-smooth curve that lies in a plane with unit normal vector n = (a, b, c). Show that the line integral 1/2 *[(bz – cy) dx + (cx – az) dy + (ay – ba) dz] equal to the plane area enclosed by C".arrow_forwardLet F = (41 + 3, sin(3t), 41² ) Find the indefinite vector integral F(t) dt = ( +C1, +C2, +C3)arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage