. (a) Each of the following functions is smooth in a simply connected region. Determinewhich of them may be written as the gradient of a scalar function, and for thosethat can, find that scalar function.i. F = yi.ii. F = Ck, where C' is a constant.iii. F = yzî + xz} + xyk.iv. F = xi + yj + zk.v. F = e* sin yi + e ¥ sin zj + e #sin yk.(b) Neither of the following functions is smooth everywhere. Nonctheless, cach canbe written as the gradient of a scalar function. Find that scalar function.i. F = 7/r², where i = xì + y}.ii. F = F/r!/2, where = xi+ yj + zk.

In order to find the scalar function we need to equate the respective partial differentiations and…

(a) Each of the following functions is smooth in a simply connected region. Determine which of them may be written as the gradient of a scalar function, and for those that can, find that scalar function. i. F = yî. ii. F = Ck, where C is a constant. iii. F = yzî + xzj + xyk. iv. F = xì + y} + zk. v. F = e* sin yi + e¯♥ sin zj + e %3D %3D * sin yk. (b) Neither of the following functions is smooth everywhcre. Nonctheless, cach can be written as the gradicnt of a scalar function. Find that scalar function. i. F = r/r², where i = xî + yj. ii. F = r/r/2, where i = xì + yì + zk. %3D

(a) Each of the following functions is smooth in a simply connected region. Determine which of them may be written as the gradient of a scalar function, and for those that can, find that scalar function. i. F = yî. ii. F = Ck, where C is a constant. iii. F = yzî + xzj + xyk. iv. F = xì + y} + zk. v. F = e* sin yi + e¯♥ sin zj + e %3D %3D * sin yk. (b) Neither of the following functions is smooth everywhcre. Nonctheless, cach can be written as the gradicnt of a scalar function. Find that scalar function. i. F = r/r², where i = xî + yj. ii. F = r/r/2, where i = xì + yì + zk. %3D

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.CR: Review Exercises

Problem 39CR: For T:R5R3 and nullity(T)=2, find rank(T).

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