Other Vector Identities - Null Vector Identities and Classiﬁcation of Vector Fields

2.5 Null Vector Identities and Classiﬁcation of Vector Fields

2.5.3 Other Vector Identities

IfUandQare scalar functions andAandBare vector functions, all dependent on the three variables (for example,x,y, andz), we can show that

∇ðUQÞ ¼Uð∇QÞ þQð∇UÞ ð2:139Þ

2.5 Null Vector Identities and Classification of Vector Fields 89

∇

^ð^UA^{Þ ¼}^U^ð^∇

^A^{Þ þ}^ð^∇U^Þ

^A ^ð2^:^140Þ

∇

^ð^A^B^{Þ ¼ A}

^ð^∇^B^{Þ þ}^ð^∇^A^Þ

^B ^ð2^:^141Þ

∇ðUAÞ ¼Uð∇AÞ þð∇UÞ A ð2:142Þ

Problems

2.1 A force is described in cylindrical coordinates asF¼ϕ^=r. Find the work performed by the force along the following paths:

(a) FromP(a,0,0) toP(a,b,c).

(b) FromP(a,0,0) toP(a,b,0), and then fromP(a,b,0) toP(a,b,c).

2.2 Determine whether ðP2

P₁

^dlbetween points p₁(0,0,0) and p₂(1,1,1) is path dependent forA¼x^y²þy^2xþz^. 2.3 A body is moved along the path shown inFigure 2.26by a forceA¼x^2y^5. The path between pointaand pointbis

a parabola described byy¼2x².

(a) Calculate the work necessary to move the body from pointato pointbalong the parabola.

(b) Calculate the work necessary to move the body from pointato pointcand then to pointb.

x y

−1 1

Figure 2.26

Surface Integrals (Closed and Open)

2.4 A volume is deﬁned in cylindrical coordinates as 1r2,π/6ϕπ/3, 1z2. Calculate the ﬂux of the vector A¼^r4zthrough the surface enclosing the given volume.

2.5 Given a surfaceS¼S1+S2defined in spherical coordinates withS1defined as 0R1;θ¼π/6; 0ϕ2πand S2defined asR¼1; 0θ π/6; 0ϕ 2π. Vector A¼R^1þθ^θ is given. Find the integral ofA

^ds^{over the}

surfaceS.

2.6 GivenA¼x^x²þy^y²þz^z², integrateA

^dsover the surface of the cube of side 1 with four of its vertices at (0,0,0), (0,0,1), (0,1,0), and (1,0,0).

2.7 The axis of a disk of radiusais in the direction of the vectork¼z^3. Vector ﬁeldA¼r^5þz^3 is given. Find the total ﬂux ofAthrough the disk.

Volume Integrals

2.8 A mass density in space is given byρ(r,z)¼r(r+ a) +z(z+d) kg/m³(in cylindrical coordinates).

(a) Calculate the total mass of a cylinder of lengthd, radiusa, centered at the origin with its axis along thezaxis.

(b) Calculate the total mass of a sphere of radiusacentered at the origin.

2.9 A right circular cone is cut off at heighth0. The radius of the small base isaand that of the large base isb(Figure 2.27).

The cone is ﬁlled with particles in a nonuniform distribution:n(r,h)¼10⁵r³+ 10³r(h–h0)². Find the total number of particles contained in the cone.

b h=0

h=h₀

h₀

Figure 2.27

2.10 Vector fieldf¼x^2xyþy^zþz^y²is defined as a volume force density (in N/m³) in a sufficiently large region in space.

This force acts on every particle of any body placed in the ﬁeld (similar to a gravitational force).

(a) A cubic body 222 m³in dimensions is placed in the ﬁeld with its center at the origin and with its sides parallel to the system of coordinates. Calculate the total force acting on the body.

(b) The same cube as in(a)is placed in the ﬁrst quadrant with one corner at the origin and with its sides parallel to the system of coordinates. Calculate the total force acting on the body.

Other Regular Integrals

2.11 The acceleration of a body is given asa¼x^ðt²2tÞ þy^3t[m/s²]. Find the velocity of the body after 5 s.

2.12 Evaluate the integralÐ

Cr²dl, wherer²¼x²+y², from the origin to the pointP(1,3) along the straight line connecting the origin toP(1,3).dlis the differential vector in Cartesian coordinates.

The Gradient

2.13 Find the derivative ofxy²+yzat (1,1,2) in the direction of the vectorx^2y^þz^2.

2.14 An atmospheric pressure ﬁeld is given asP(x,y,z)¼(x– 2)²+ (y– 2)²+ (z+ 1)², where thex–yplane is parallel to the surface of the ocean and thezdirection is vertical. Find:

(a) The magnitude and direction of the pressure gradient.

(b) The derivative of the pressure in the vertical direction.

2.15 The scalar ﬁeldf(r,ϕ,z)¼rcos²ϕ+ zsinϕis given. Calculate:

(a) The gradient off(r,ϕ,z) in cylindrical coordinates.

(b) The gradient off(r,ϕ,z) in Cartesian coordinates.

2.16 Find the unit vector normal to the following planes:

(a) z¼–5x– 3y.

(b) 4x– 3y+z+ 5¼0.

Show by explicit derivation that the result obtained is in fact normal to the plane.

2.17 Find the unit vector normal to the following surfaces:

(a) z¼–3xy–yz.

(b) x¼z²+y². (c) z²+ y²+x²¼8.

Problems 91

The Divergence

2.18 Calculate the divergence of the following vector ﬁelds:

(a) A¼x^x²þy^1z^y²: (b) B¼r^2z²þϕ^5rz^3r²: (c) C¼x^ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x²þz²

p þy^ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x²þy²

p :

2.19 Find the divergence ofA¼x^x²þy^y²þz^z²at (1,–1,2).

2.20 Find the divergence ofA¼r^2rcosϕϕ^rsinϕþz^4zat 2ð ;90,1Þ:

2.21 Find the divergence ofA¼0:2R³ϕsin²θR^ þθ^þϕ^

at 2ð ;30;90Þ:

The Divergence Theorem

2.22 Verify the divergence theorem forA¼x^4zy^2y²z^2z²for the region bounded byx²+y²¼9 andz¼–2,z¼2 by evaluating the volume and surface integrals.

2.23 A vector ﬁeld is given as A(R)¼R, whereRis the position vector of a point in space. Show that the divergence theorem applies to the vectorAfor a sphere of radiusa.

2.24 GivenA¼x^x²þy^y²þ^zz²:

(a) IntegrateA

^dsover the surface of the cube of side 1 with four of its vertices at (0,0,0), (0,0,1), (0,1,0), and (1,0,0) (seeProblem 2.6).

(b) Integrate∇

^Aover the volume of the cube in(a)and show that the two results are the same.

The Curl

2.25 Calculate the curl of the following three vectors:

(a) A¼x^x²þy^1z^y²: (b) B¼r^2z²þϕ^5rz^3r²: (c) C¼x^ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x²þz²

p þy^ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x²þy²

p :

2.26 A ﬂuid ﬂows in a circular pattern with the velocity vectorv¼ϕ^=r:

(a) Sketch the vector ﬁeldv.

(b) Calculate the curl of the vector ﬁeld.

2.27 A vector ﬁeldA¼y^3xcosðωtþ50zÞis given.

(a) What is the curl ofA?

(b) Is this a conservative ﬁeld?

2.28 Find∇Afor:

(a) A¼x^y²þy^2ðxþ1Þyzz^ðxþ1Þz²: (b) A¼r^2rcosϕϕ^4rsinϕþz^3:

Stokes’Theorem

2.29 Verify Stokes’theorem forA¼x^ð2xyÞ y^2yz²z^2zy²on the upper half-surface of the spherex²+y²+z²¼4 above thexyplane. The contour bounding the surface is the rim of the half-sphere.

2.30 Vector ﬁeldF¼x^3yþy^ð52xÞ þz^ðz²2Þis given. Find:

(a) The divergence ofF.

(b) The curl ofF.

2.31 Vector ﬁeldF¼x^yþy^zþz^xis given. Find the total ﬂux of∇Fthrough a triangular surface given by three points P1(a,0,0),P2(0,0,b), andP3(0,c,0).

The Helmholtz Theorem and Vector Identities

2.32 The following vector operations are given:

1.∇

⁽^∇ϕ⁾ ^{2. (}^∇

^∇⁾^ϕ

3. (∇∇)ϕ 4.∇(∇ϕ)

5.∇

⁽^∇^A) ^{6. (}^∇

^∇⁾^A

7. (∇∇)A 8.∇(∇A)

9.∇

⁽^ϕ^∇^A) ^10.^ϕ⁽^∇^A)

11.∇(∇A) 12.∇(∇

^A)

whereAis an arbitrary vector ﬁeld andϕan arbitrary scalar ﬁeld.

(a) Which of the operations are valid?

(b) Evaluate explicitly those that are valid (in Cartesian coordinates).

2.33 Calculate the Laplacian for the following scalar ﬁelds:

(a) p¼(x– 2)²(y– 2)²(z+ 1)². (b) p ¼5rcosϕ+ 3zr².

2.34 Calculate the Laplacian for the following vector ﬁelds:

(a) A¼x^3yþy^ð52xÞ þz^ðz²2Þ:

(b) A¼r^2rcosϕϕ^4rsinϕþz^3:

2.35 Show that ifFis a conservative ﬁeld, then∇²F¼∇(∇

F). Use cylindrical coordinates.

2.36 Given the scalar fieldf(x,y,z)¼2x²+yand the vector fieldR¼x^xþy^yþz^z, find:

(a) The gradient off.

(b) The divergence offR.

(d) The vector Laplacian ofR.

(e) The curl offR.

2.37 A vector ﬁeldA¼x^5xþy^2yþz^1. What type of ﬁeld is this according to the Helmholtz theorem?

2.38 A vector ﬁeld A¼R^ϕR²þθ^Rsinθ is given in spherical coordinates. What type of ﬁeld is this according to the Helmholtz theorem?

2.39 The following vector ﬁelds are given:

(1) A¼x^xþy^y:

(2) B¼ϕ^cosϕþr^cosϕ:

(3) C¼x^yþz^y:

(4) D¼R^sinθþθ^5Rþϕ^Rsinθ:

(5) E¼R^k:

Problems 93

(a) Which of the ﬁelds are solenoidal?

(b) Which of the ﬁelds are irrotational?

2.40 Show by direct derivation of the products that the following holds:

∇ð∇AÞ ¼∇ ∇^ð

^A^Þ^∇²^A^:

Coulomb ’ s Law and the Electric Field

3

I looked, and lo, a stormy wind came sweeping out of the north—a huge cloud and flashing fire, surrounded by a radiance; and in the center of it, in the center of the fire, a gleam as of amber.

—Ezekiel 1:4

Dalam dokumen Nathan Ida Third Edition - Engineering Electromagnetics (Halaman 116-122)