Begin2.DVI
8-13. A force field F is said to be conservative if it is derivable from a scalar potential function V such that F=±grad V. O ...
8-18. For r =xˆe 1 +yeˆ 2 +zˆe 3 and r=|r |show (i) curl rnr = 0 (ii) div r = 3 (iii) curl r = 0 (iv) div rnr = (n+ 3)r ...
8-25. Use line integrals to find the area enclosed by the given curves. (i) The ellipse, x=acos t y =bsin t, 0 ≤t≤ 2 π. (ii) Th ...
8-31. Let r denote a position vector to a general point on a closed surface S, which encloses a volume V. Evaluate the surface ...
8-37. Let F=xˆe 1 +yˆe 2 +zˆe 3 and evaluate the surface integral I= ∫∫ S F·dS , where Sis the surface enclosing the volume ...
8-42. In cylindrical coordinates (r, θ, z ),show that curl F=∇× F =^1 r ∣∣ ∣∣ ∣∣ ˆer rˆeθ eˆz ∂ ∂r ∂ ∂θ ∂ ∂z Fr rF θ Fz ∣∣ ∣∣ ...
8-51. Show in a general orthogonal coordinate system div (grad v×grad w) = 0 8-52. For ˆe 1 ,ˆe 2 ,ˆe 3 independent orthogonal ...
8-54. Let E 1 ,E 2 ,E 3 and E^1 ,E^2 ,E^3 be a system of reciprocal basis. (See previous problem). (a) If A=A^1 E 1 +A^ ...
Chapter9 Applications of Vectors The use of vectors in mathematics, physics, engineering and the sciences is extensive. The appl ...
d 1 = √ (x−x 1 )^2 + (y−y 1 )^2 + (z−z 1 )^2 d 2 = √ (x−x 2 )^2 + (y−y 2 )^2 + (z−z 2 )^2 .. . dn= √ (x−xn)^2 + (y−yn)^2 + (z−zn ...
known value at i occurs. This approximation method is an interpolation method associated with the given data values and is known ...
Define the unit vectors ˆeA,eˆB,ˆeCfrom the center of the unit sphere to the points A, B, C on the surface of the sphere and obs ...
Using similar arguments associated with the representation of cos αand cos β, one can show cos b= cos ccos a+ sin csin acos β co ...
Velocity and Acceleration in Polar Coordinates In polar coordinates (r, θ)one can employ the orthogonal unit vectors ˆer= cosθˆe ...
Here the radial component of acceleration is ( ̈r−r(θ ̇)^2 )and the transverse component of acceleration or tangential component ...
since ˆerchanges with time, but ˆez=ˆe 3 remains constant. From equation (9.27) one can calculate the derivatives dˆer dt =−sin ...
Velocity and Acceleration in Spherical Coordinates Upon changing to spherical^3 coordinates (ρ, θ, φ ) the transformation equati ...
v =dr dt =ρdˆeρ dt +dρ dt ˆeρ =ρ ( dθ dt ˆeθ+ sin θdφ dt ˆeφ ) +dρ dt ˆeρ = ̇ρˆeρ+ρθ ̇ˆeθ+ρφ ̇sin θˆeφ (9 .38) Here vρ= ̇ρis t ...
or to solve for one quantity φ=φ(x, y, z ),then it should be obvious that it would be easier to solve for the one quantity φand ...
denote two points in the simply connected region Rof the vector field F . The work done can be expressed by performing a line i ...
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