Find thefollowing:a.) Find the surface area of the portionof the cylinder x²+y²=4 above the xyplane and below the cylinder z=4-y²b.) Determine if there is a vector field Fsuch that curl F =< x³, 2y, z>. Justify theanswer briefly.

Since you have posted multiple questions we are supposed to give solution of first one only kindly…

Answered: Find the following: a.) Find the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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H.W: Select the divergence of F(x, y) = i+ (2x – 3y)j. 1 1 (a) 3, (b) (c) (d) -2. +2, y? y?
WHICH ANSWER IS CORRECT?
You are given the scalar field = -7y. a) b) c) d) Calculate Vo. Vo =( Calculate V at the point p = (-1, -2,-2). Vo (-1, -2,-2) =( ). Calculate the unit vector in the direction of n = ( 1, 3, 1 ). n = ( ). (Enter your answers to 3d.p.) Calculate the directional derivative Von, of the scalar field , in the direction of n, at the point p. (V¢·n)(−1,−2, –2) = (Enter your answer to 3d.p.)
Consider the graph of an ellipse C, oriented counterclockwise. Graphs of the vector fields F,, F,, F, and F, along the curve C are given (see figures). Along C, F, and F, are tangent to C, and F, and F, are normal to C. Determine whether the following integrals are positive, negative, or equal to 0. b. F, · T ds JF T ds а. ·T ds с. C d. F -T ds F -n ds f. F,•n ds е. g. F•n ds h. Fn ds Vectors of F1 along C Vectors of F2 along C Vectors of F3 along C Vectors of F4 along C
Show that if F = xi+yj + zk, then VXF=0. The curl of a vector field F = Mi+Nj + Pk is as follows. ap ƏN ?M ap dy dz dz ?x Use the formula above to find the curl of F=xi+yj + zk. curl F = VXF= VxF = V x (xi+yj+zk) dz dy = - i+ dy dz = i+ İ+ dx dz dz ?x + k j+ dy ax ?x ду ƏN ax k ƏM ду
a) Draw 4 pictures showing all possible combinations of a vector field and a curve b) Compute the circulation and the flux
In Cartesian coordinates, a vector field takes the form F= 2xzi+ 2yzj+ (X2+y²)k a. State whether F is conservative, and give a one-sentence justification for your statement b. Calculate the line integral of F along a straight-line path starting at the origin and ending at the point (a, b, c). This path has the parametric representation x=at, y=bt, z=ct (0sts1). c. Given that the point (a, b, c) could be anywhere, use your answer to part (b) to find the scalar potential function U(x, y, z) corresponding to F, such that F=-VU.

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Find the following: a.) Find the surface area of the portion of the cylinder x²+y²=4 above the xy plane and below the cylinder z=4-y² b.) Determine if there is a vector field F such that curl F =< x³, 2y, z>. Justify the answer briefly.