PROBLEMS Section 3-1: Vector Algebra *3.1 Vector A starts at point (1, -1, -3) and ends at point (2, -1, 0). Find a unit vector in the direction of A ฬ2 โ ๐ ฬ3 + ๐ฬ, ๐ฉ = ๐ ฬ2 โ ๐ ฬ + ๐ฬ3, ๐๐๐ ๐ถ = ๐ ฬ4 + ๐ฬ 2, โ๐ฬ2, show that C is 3.2 Given vectors ๐จ = ๐ perpendicular to both A and B. *3.3 In Cartesian coordinates, the three corners of a triangle are ๐1 = (0, 4, 4), ๐2 = (4, โ4, 4) ๐๐๐ ๐3 = (2, 2, โ4), Find the area of the triangle. ฬ2 โ ๐ ฬ๐ + ๐ฬ1 ๐๐๐ ๐ฉ = ๐ ฬ๐ต๐ + ๐ ฬ2 + ๐ฬ๐ต๐ : 3.4 Given ๐จ = ๐ (a) Find ๐ต๐ ๐๐๐ ๐ต๐ if A is parallel to B. (b) Find a relation between ๐ต๐ ๐๐๐ ๐ต๐ if A is perpendicular to B. ฬ+๐ ฬ2 + ๐ฬ3, ๐ฉ = ๐ฅฬ๐ โ ๐ ฬ4, ๐๐๐ ๐ถ = ๐ ฬ2 โ ๐ฬ4, find 3.5 Given vectors ๐จ = ๐ ฬ (a) A and ๐, (b) the component of B along C, (c) ฮธ๐ด๐ถ, (d) ๐ ๐ฑ ๐, (e) ๐ · (๐ฉ ๐ ๐ช), (f) ๐จ ๐ (๐ฉ ๐ ๐ช), (g) ฬ ๐ ๐ฅ ๐ฉ, ๐๐๐ ฬ) · ๐ฬ. (h) (๐จ ๐ ๐ ฬ๐ โ ๐ ฬ + ๐ฬ3 ๐๐๐ ๐ฉ = ๐ ฬ3 โ ๐ฬ2, find a vector C whose magnitude is 9 and 3.6 Given vectors ๐จ = ๐ whose direction is perpendicular to both A and B. ฬ(๐ฅ + 2๐ฆ) โ ๐ ฬ(๐ฆ + 3๐ง) + ๐งฬ (3๐ฅ โ ๐ฆ), determine a unit vector parallel to A at *3.7 Given ๐จ = ๐ point ๐ = (1, โ1, 2). 3.8 By expansion in Cartesian coordinates, prove: (a) The relation for the scalar triple product given by Eq. (3.29). (b) The relation for the vector triple product given by Eq. (3.33). 3.9 Find an expression for the unit vector directed toward the origin from an arbitrary point on the line described by x=1 and z = -3. 3.10 Find an expression for the unit vector directed toward the point P located on the z-axis at a height h above the x-y plane from an arbitrary point ๐ = (๐ฅ, ๐ฆ, , โ5) in the plane z= -5. 3.11 Find a unit vector parallel to either direction of the line described by 2๐ฅ + ๐ง = 4. 3.12 Two lines in the x-y plane are described by the following expressions: Line 1 ๐ฅ + 2๐ฆ = โ6 Line 2 3๐ฅ + 4๐ฆ = 8 Use vector algebra to find the smaller angle between the lines at their intersection point. *3.13 A given line is described by ๐ฅ + 2๐ฆ = 4 Vector A starts at the origin and ends at point P on the line such that A is orthogonal to the line. Find an expression for A. 3.14 how that, given to two vectors A and B, (a) The vector C defined as the vector component of B in the direction of A is given by ๐จ(๐ฉ · ๐จ) ฬ (๐ฉ · ๐) = ๐ช=๐ |๐จ|2 ฬ is the unit vector of A. Where ๐ (b) The vector D defined as the vector component of B perpendicular to A is given by ๐จ(๐ฉ · ๐จ) ๐ซ=๐ฉโ . |๐จ|๐ *3.15 A certain plane is described by 2๐ฅ + 3๐ฆ + 4๐ง = 16, Find the unit vector normal to the surface in the direction away from the origin. ฬ(๐ง โ 3๐ฆ) + ๐ ฬ(2๐ฅ โ 3๐ง) โ ๐ฬ(๐ฅ + ๐ฆ), find a unit vector parallel to B at point 3.16 Given ๐ฉ = ๐ P = (1, 0, -1). 3.17 Find a vector G whose magnitude is 4 and whose direction is perpendicular to both vectors ฬ+๐ ฬ ๐ โ ๐ฬ 2 and ๐ญ = ๐ ฬ 3 โ ๐ฬ 6. E and F, where ๐ฌ = ๐ 3.18 A given line is described by the equation: ๐ฆ = ๐ฅ โ 1. Vector A starts at point ๐1 = (0, 2) and ends at point ๐2 on the line, at which A is orthogonal to the line. Find an expression for A. 3.19 Vector field E is given by ฬ 5๐ cos ๐ โ ๐ฝ ฬ12 ฬ ๐ฌ=๐น ๐ sin ๐ cos ๐ + ๐3 sin ๐. Determine the component of E tangential to the spherical surface R = 2 at point P = (2, 300, 600). 3.20 When sketching of demonstrating the spatial variation of vector field, we often use arrows, as in Fig. P3.20, wherein the length of the arrow is made to be proportional to the strength of the field and the direction of the arrow is the same as that of the fieldโs. The sketch shown in Fig.P3.20, which represents the vector field ๐ฌ = ๐ฬ๐, consists of arrows pointing radially away from the origin and their lengths increasing linearly in proportion to their distance away from the origin. Using this arrow representation, sketch each of the following vector fields: ฬ๐ฆ (a) ๐ฌ1 = โ๐ ฬ๐ (b) ๐ฌ2 = ๐ ฬ๐ฅ + ๐ ฬ๐ (c) ๐ฌ3 = ๐ ฬ๐ฅ + ๐ ฬ2๐ฆ (d) ๐ฌ4 = ๐ ฬ (e) ๐ฌ๐ = ๐๐ (f) ๐ฌ6 = ะณฬ sin ๐ 3.21 Use arrows to sketch each of the following vector fields: ฬ๐ฆ (a) ๐ฌ1 = ๐ฅฬ๐ฅ โ ๐ ฬ (b) ๐ฌ2 = โ ๐ ฬ (1โ๐ฅ ) (c) ๐ฌ๐ = ๐ (d) ๐ฌ4 = ะณฬ cos ๐ Section 3-2 and 3-3: Coordinate Systems *3.22 Convert the coordinates of the following points from Cartesian to cylindrical and spherical coordinates: (a) ๐1 = (1, 2, 0) (b) ๐2 = (0, 0, 2) (c) ๐3 = (1, 1, 3) (d) ๐4 = (โ2, 2, โ2) 3.23 Convert the coordinates of the following points from cylindrical to Cartesian coordinates: (a) ๐1 = (2, ๐โ4 , โ3) (b) ๐2 = (3, 0, โ2) (c) ๐3 = (4, ๐, 5) *3.24 Convert the coordinates of the following points from spherical to cylindrical coordinates: (a) ๐1 = (5, 0, 0) (b) ๐2 = (5, 0, ๐ ) (c) ๐3 = (3, ๐โ2, 0) 3.25 Use the appropriate expression for the differential surface area ds to determine the area of each of the following surface: (a) ๐ = 3; 0 โค ๐ โค ๐โ3; โ2 โค ๐ง โค 2 (b) 2 โค ๐ โค 5; ๐โ2 โค ๐ โค ๐; ๐ง = 0 (c) 2 โค ๐ โค 5; ๐ = ๐โ4 ; โ2 โค ๐ง โค 2 (d) ๐ = 2; 0 โค ๐ โค ๐โ3 ; 0 โค ๐ โค ๐ (e) 0 โค ๐ โค 5; ๐ = ๐โ3; 0 โค ๐ โค 2๐ Also sketch the outline of each surface. *3.26 Find the volumes described by (a) 2 โค ๐ โค 5; ๐โ2 โค ๐ โค ๐; 0 โค ๐ง โค 2, ๐๐๐ (b) 0 โค ๐ โค 5; 0 โค ๐ โค ๐โ3 ; 0 โค ๐ โค 2๐. Also sketch the outline of each volume. 3.27 A section of a sphere is described by 0 โค ๐ โค 2, 0 โค ๐ โค 900 , ๐๐๐ 300 โค ๐ โค 900 . Find the following: (a) The surface area of the spherical section. (b) The enclosed volume. Also sketch the outline of the section. *3.28 A vector field is given in cylindrical coordinates by ฬ ๐ sin ๐ + ๐ฬ๐ง 2 . ๐ฌ = ๐ฬ๐ cos ๐ + ๐ Point ๐ = (2, ๐, 3) is located on the surface of the cylinder described by ๐ = 2. At point P, find: (a) The vector component of E perpendicular to the cylinder. (b) The vector component of E tangential to the cylinder 3.29 At given point in space, vectors A and B are given in spherical coordinates by ฬ, ฬ4 + ๐ฝ ฬ2 + ๐ ๐จ=๐น ฬ 3. ฬ2 + ๐ ๐ฉ = โ๐น Find: (a) The scalar component, or projection, of B in the direction of A. (b) The vector component of B in the direction of A. (c) The vector component of B perpendicular to A. *3.30 Given vectors ฬ (2๐ + 4 sin ๐) + ๐ฬ(๐ โ 2๐ง), ๐จ = ๐ฬ(cos ๐ + 3๐ง) โ ๐ ๐ฉ = โ๐ฬ sin ๐ + ๐ฬ cos ๐, Find (a) ๐๐ด๐ต ๐๐ก (2, ๐โ2, 0). (b) A unit vector perpendicular to both A and B at (2, ๐โ3, 1). 3.31 Find the distance between the following pairs of points: (a) ๐1 = (1, 2, 3)๐๐๐ ๐2 = (โ2, โ3, โ2) in Cartesian coordinates. (b) ๐3 = (1, ๐โ4, 3)๐๐๐ ๐4 = (3, ๐โ4, 4) in cylindrical coordinates, (c) ๐5 = (4, ๐โ2 , 0) ๐๐๐ ๐6 = (3, ๐, 0) in spherical coordinates. *3.32 Determine the distance between the following pairs of points: (a) ๐1 = (1, 1, 2 )๐๐๐ ๐2 = (0, 2, 3) (b) ๐3 = (2, ๐โ3, 1) ๐๐๐ ๐4 = (4, ๐โ2, 3) (c) ๐5 = (3, ๐, ๐โ2)๐๐๐ ๐6 = (4, ๐โ2, ๐) 3.33 Transform the vector ฬ sin ๐ ฬ ๐ ๐๐๐ ๐ฝ cos ๐ + ๐ฝ ฬ ๐๐๐ 2 ๐ โ ๐ ๐จ=๐น into cylindrical coordinates and then evaluate it at ๐ = (2, ๐โ2, ๐โ2). 3.34 Transform the following vectors into cylindrical coordinates and then evaluate them at the indicated points: ฬ(๐ฅ + ๐ฆ)๐๐ก ๐1 = (1, 2, 3) (a) ๐จ = ๐ ฬ (๐ฆ โ ๐ฅ ) + ๐ ฬ(๐ฅ โ ๐ฆ)๐๐ก ๐2 = (1, 0, 2) (b) ๐ฉ = ๐ ฬ๐ฆ 2 ๐ ฬ๐ฅ 2 ๐ (c) ๐ถ = (๐ฅ 2+๐ฆ 2 ) โ (๐ฅ 2+๐ฆ 2) + ๐ฬ4 ๐๐ก ๐3 = (1, โ1, 2) ฬ ๐๐๐ 2 ๐ ๐๐ก ๐4 = (2, ๐โ2, ๐โ4). ฬ sin ๐ + ๐ฝ ฬ cos ๐ + ๐ (d) ๐ซ = ๐น ฬ ๐ ๐๐๐ ๐ฝ ๐๐ก ๐5 = (3, ๐โ2, ๐) ฬ cos ๐ + ๐ฝ ฬ sin ๐ + ๐ (e) ๐ฌ = ๐น *3.35 Transform the following vectors into spherical coordinates and then evaluate them at the indicated points: ฬ ๐2 + ๐ ฬ๐๐ + ๐ฬ4 ๐๐ก ๐1 = (1, โ1, 2) (a) ๐จ = ๐ ๐ ฬ(๐ฅ + ๐ฆ ๐ + ๐ง ๐ ) โ ๐ฬ(๐ฅ ๐ + ๐ฆ ๐ ) ๐๐ก ๐2 = (โ1, 0, 2) (b) ๐ฉ = ๐ ฬ sin ๐ + ๐ฬ cos ๐ sin ๐ ๐๐ก ๐3 = (2, ๐ , 2) (c) ๐ช = ๐ฬ cos ๐ โ ๐ 4 ฬ๐ฅ 2 โ(๐ฅ 2 + ๐ฆ 2 ) + ๐ฬ4 ๐๐ก ๐4 = (1, โ1, 2) (d) ๐ซ = ฬ ๐๐ฆ 2 โ ( ๐ฅ 2 + ๐ฆ 2 ) โ ๐ Section 3-4 to 3-7: Gradient Divergence, and Curl Operators 3.36 Find the gradient of the following scalar functions: (a) ๐ = 3โ(๐ฅ 2 + ๐ง 2 ) (b) ๐ = ๐ฅ๐ฆ 2 ๐ง 4 (c) ๐ = ๐ง cos ๐ โ(1 + ๐ 2 ) (d) ๐ = ๐ โ๐ sin ๐ (e) ๐ = 4๐ฅ 2 ๐ โ๐ง + ๐ฆ 3 (f) ๐ = ๐ 2 ๐๐๐ 2 ๐ (g) ๐ = ๐ cos ๐ sin ๐ 3.37 For each of the following scalar fields, obtain an analytical solution for โT and generate a corresponding arrow representation. = 10 + ๐ฅ, ๐๐๐ โ 10 โค ๐ฅ โค 10 = ๐ฅ 2 , ๐๐๐ โ 10 โค ๐ฅ โค 10 = 100 + ๐ฅ๐ฆ, ๐๐๐ โ 10 โค ๐ฅ โค 10 = ๐ฅ 2 ๐ฆ 2 , ๐๐๐ โ 10 โค ๐ฅ, ๐ฆ โค 10 = 20 + ๐ฅ + ๐ฆ, ๐๐๐ โ 10 โค ๐ฅ, ๐ฆ โค 10 = 1 + sin(๐๐ฅโ3), ๐๐๐ โ 10 โค ๐ฅ โค 10 = 1 + cos(๐๐ฅโ3), ๐๐๐ โ 10 โค ๐ฅ โค 10 0 โค ๐ โค 10 } (h) ๐ = 15 + ๐ cos ๐ , ๐๐๐ { 0 โค ๐ โค 2๐. 0 โค ๐ โค 10 } (i) ๐ = 15 + ๐๐๐๐ 2 ๐, ๐๐๐ { 0 โค ๐ โค 2๐. 3.38 The gradient of a scalar function T is given by (a) (b) (c) (d) (e) (f) (g) ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ป๐ = ๐ฬ๐ โ2๐ง . If T = 10 at z = 0, find T (z). 3.39 Follow a procedure similar to that leading to Eq. (3.82) to derive the expression given by Eq. (3.83) for โ in spherical coordinates. 3.40 For the scalar function ๐ = ๐ฅ๐ฆ 2 โ ๐ง 2 , determine its directional derivative along the direction ฬโ๐ ฬ๐ง) and then evaluate it at ๐ = (1, โ1, 4). of vector ๐จ = (๐ ฬ๐ฅ โ ๐ ฬ๐ฆ along the segment P1 to P2 of the circular path 3.41 Evaluate the line integral of ๐ฌ = ๐ shown in Fig. P3.41. 1 3.42 For the scalar function ๐ = 2 ๐ โ๐ โ5 cos ๐, determine its directional derivative along the radial direction ๐ฬ and then evaluate it at ๐ = (2, ๐โ4, 3). 1 *3.43 For the scalar function ๐ = ๐ ๐ ๐๐2 ๐, determine its directional derivative along the range ฬ and then evaluate it at ๐ = (5, ๐โ4, ๐โ2). direction ๐น 3.44 Each of the following vector fields is displayed in Fig.P3.44 in the form of a vector representation. Determine โ·A analytically and then compare the result with your expectations on the basis of the displayed pattern. (a) (b) (c) (d) (e) (f) (g) ฬ cos ๐ฅ sin ๐ฆ + ๐ ฬ sin ๐ฅ cos ๐ฆ, ๐๐๐ โ ๐ โค ๐ฅ, ๐ฆ โค ๐ ๐จ = โ๐ ฬ sin 2๐ฆ + ๐ ฬ cos 2๐ฅ, ๐๐๐ โ ๐ โค ๐ฅ, ๐ฆ โค ๐ ๐จ = โ๐ 2 ฬ๐ฅ๐ฆ + ๐ ฬ๐ฆ , ๐๐๐ โ 10 โค ๐ฅ, ๐ฆ โค 10 ๐จ = โ๐ ฬ cos ๐ฅ + ๐ ฬ sin ๐ฆ, ๐๐๐ โ ๐ โค ๐ฅ, ๐ฆ โค ๐ ๐จ = โ๐ ฬ๐ฅ, ๐๐๐ โ 10 โค ๐ฅ โค 10 ๐จ=๐ ฬ๐ฅ๐ฆ 2 , ๐๐๐ โ 10 โค ๐ฅ, ๐ฆ โค 10 ๐จ=๐ ฬ๐ฅ๐ฆ ๐ + ๐ ฬ๐ฅ 2 ๐ฆ, ๐๐๐ โ 10 โค ๐ฅ, ๐ฆ โค 10 ๐จ=๐ ๐๐ฅ ๐๐ฆ ฬ sin ( ) + ๐ ฬ sin ( ) , โ10 โค ๐ฅ, ๐ฆ โค 10 (h) ๐จ = ๐ 10 10 ฬ ๐ cos ๐ , ๐๐๐ { 0 โค ๐ โค 10 } (i) ๐จ = ๐ฬ๐ + ๐ 0 โค ๐ โค 2๐. 0 โค ๐ โค 10 } (j) ๐จ = ๐ฬ๐ ๐ + ๐ฬ๐ 2 sin ๐ ๐๐๐ { 0 โค ๐ โค 2๐. ฬ ; (๐) the 3.45 Vector field E is characterized by the following properties: (a) E points along ๐น magnitude of E is a function of only the distance from the origin; (c) E vanishes at the origin; and (d) โ·E =12, everywhere. Find an expression for E that satisfies these properties. ฬ๐ฅ๐ง โ ๐ ฬ๐ฆ๐ง ๐ โ ๐ฬ๐ฅ๐ฆ, verify the divergence theorem by *3.46 For the vector field ๐ฌ = ๐ computing: (a) The total outward flux flowing through the surface of a cube centered at the origin and with sides equal to 2 units each and parallel to the Cartesian axes. (b) The integral of โ·E over the cubeโs volume. 3.47 For the vector field ๐ฌ = ๐ฬ10โ๐ โ ๐ฬ3๐ง, verify the divergence theorem for the cylindrical region enclosed by ๐ = 2, ๐ง = 0 ๐๐๐ ๐ง = 4, *3.48 A vector field ๐ซ = ๐ฬ๐ 3 exists in the region between two concentric cylindrical surfaces defined by r = 1 and r = 2, with both cylinders extending between z = 0 and z = 5. Verify the divergence theorem by evaluating (a) โฎ๐ ๐ซ · ๐๐, (b) โซ๐ ๐ป · ๐ซ ๐๐. ฬ ๐๐ ๐ , evaluate both sides of the divergence theorem for the region 3.49 For the vector field ๐ซ = ๐น enclosed between the spherical shells defined by R = 1 and R = 2. ฬ๐ฅ๐ฆ โ ๐ ฬ(๐ฅ 2 + 2๐ฆ 2 ), calculate 3.50 For the vector field ๐ฌ = ๐ (a) โฎ๐ ๐ฌ · ๐๐ฐ around the triangular contour shown in P3.50(a). (b) โซ๐ (๐ป๐ ๐ฌ) · ๐๐ over the area of the triangle. 3.51 Repeat Problem 3.50 for the contour shown in 3.50(b). *3.52 Verify Stokeโs theorem for the vector field ฬ sin ๐) ๐ฉ = (๐ฬ๐ cos ๐ + ๐ By evaluating (a) โฎ๐ ๐ฉ · ๐๐ฐ over the semicircular contour shown in Fig.P3.52(a). (b) โซ๐ (๐ป๐ ๐ฉ) · ๐๐ over the surface of the semicircle. 3.53 Repeat Problem 3.52 for the contour shown in Fig.P3.52(b) ฬ sin ๐ by evaluating it on the ฬ cos ๐ + ๐ 3.54 Verify Stokeโs theorem for the vector field ๐จ = ๐น hemisphere of unit radius. ฬ sin ๐) by evaluating: 3.55 Verify Stokeโs theorem for the vector field ๐ฉ = (๐ฬ cos ๐ + ๐ (a) โฎ๐ ๐ฉ · ๐๐ต over the path comprising a quarter section of a circle, as shown in Fig.P3.55, and (b) โซ๐ (๐ป๐ ๐ฉ) · ๐๐ over the surface of the quarter section. *3.56 Determine if each of the following vector fields is solenoidal, conservative, or both: (a) (b) (c) (d) ฬ๐ฅ 2 โ ๐ ฬ2๐ฅ๐ฆ ๐จ=๐ ๐ ฬ๐ฅ โ ๐ ฬ๐ฆ ๐ + ๐ฬ๐๐ ๐ฉ=๐ ๐ช = ๐ฬ (sin ๐)โ๐ 2 + ๐ฬ (cos ๐)โ๐ ๐ ฬ โ๐ ๐ซ=๐น ๐ (e) ๐ฌ = ๐ฬ (3 โ 1+๐) + ๐ฬ๐ง ฬ๐ฆ + ๐ ฬ๐ฅ )โ(๐ฅ 2 + ๐ฆ 2 ) (f) ๐ญ = (๐ ฬ (๐ฅ 2 + ๐ง 2 ) โ ๐ ฬ(๐ฆ 2 + ๐ฅ 2 ) โ ๐ฬ(๐ฆ 2 + ๐ง 2 ) (g) ๐ฎ = ๐ ฬ (๐ ๐ โ๐ ) (h) ๐ฏ = ๐น 3.57 Find the Laplacian of the following scalar functions: (a) ๐ = 4๐ฅ๐ฆ 2 ๐ง 3 (b) ๐ = ๐ฅ๐ฆ + ๐ฆ๐ง + ๐ง๐ฅ (c) ๐ = 3โ(๐ฅ 2 + ๐ฆ 2 ) (d) ๐ = 5๐ โ๐ cos ๐ (e) ๐ = 10๐ โ๐ sin ๐ 3.58 Find the Laplacian of the following scalar functions: (a) ๐1 = 10๐ 3 sin 2๐ (b) ๐2 = (2โ๐ 2 ) cos ๐ sin ๐