Finding Work in a Conservative Force Field
In exercise 19-22, (a) show the
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Chapter 15 Solutions
Calculus: Early Transcendental Functions
- +y -n CI Č2 FIGURE Q3(b)arrow_forwardExercise III Let (a) o = x²y +xż and (b) ó = x² + y² + z?. Then, respectively attempt to find the directional derivative at • (1,2, 1) in the direction of the vector (2i - 3j+ 4k). • (3,0, 1) in the direction of the vector (i- 3j + 2k).arrow_forwardFind the gradient vector field of f(z, y) = r'y* Question Help: OVideo Submit Questionarrow_forward
- 10) Find the work done by the force field F(x,y,z) = zi + x j + y k in moving a particle from the point (3, 0, 0) to the point (0,TT/2,3) along (a) straight line (b) the helix x = 3 cost, y = t, z = 3 sintarrow_forwardSketch and describe the vector field F (x, y) = (-y,2x)arrow_forward= Calculate the flux of the vector field F(x, y, z) = (5x + 9)ỉ through a disk of radius 3 centered at the origin in the yz-plane, oriented in the negative x-direction. Flux =arrow_forward
- a) Find the values of a,b,c which make f(x,y,z) = sin (xay°z©) a potential function for the vector field F(x.y,z)= 4 x³y'z? cos (x*y z7) i+ 7x*y®z7 cos (x*y'z?) j + 7x*y?z® cos (x*y'z7) k a= b= c= b) Find the work of F along the curve r(t)=(2t, e t(t – 1), e tt – 1)) for 0sts1. Do not approximate your answer. For example, leave sin (e ) as that, not 0.41. Work = F•dr =arrow_forward(b) Find the work done by force field F = 3x? î + y²j on a particle when it moves from (0, 0) to (-T, 0) along the curves Cl and C2 in Figure Q3 (b) by solving S. F dr. Based on your calculation, judge whether the force, F is conservative or non- conservative and give your explanation. | -TT C1 0, C2 FIGURE Q3(b)arrow_forwardVector 1 A) Prove that: V x (FxG) = (GV)F- (FV)G+ F(VG) - G(V.F). B) Find all of the second derivatives for f(x. y) = (3xy² + 2xy + x²) In: +y² Brie differente directia derivatives with stable gnh an eations. Find the direcional avati 220. vector in t direction of ere E) Evaluat the trde integral I need answer Only branch A (a) dx dy= (b) dyd = 24 GEL direct رسالت +vana is the unitarrow_forward
- Prove that F = (2n -)-)+ick is a conservative force field. Find the scalar 4.A. potential for Fand the work done in moving an object in this field from (1,-2,1) to (3,1,4). %3D Find the value of a, b, c so that the directional derivative of 4B. O = axy-byz+czx' has a maximum of magnitude 64 in the direction of z-axis.arrow_forwarda) Find the values of a,b,c which make f(x,y,z) = sin (xªy°z©) a potential function for the vector field F(x,y,z)= 2 x'y®z4 cos (x?y®z4) i+ 5x²y^z* cos (x?y5z4) j + 4 x?y5z³ cos (x²y®z4) k a= b= C= b) Find the work of F along the curve r(t)=(2t, e (t - 1), e t(t – 1) for 0 sts1. Do not approximate your answer. For example, leave sin (e ) as that, not 0.41. Work = F•dr =arrow_forwardWork by a constant force Show that the work done by a con- stant force field F = ai + bj + ck in moving a particle along any path from A to B is W = F•AB.arrow_forward
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