Begin2.DVI
where E=∂r ∂u ·∂r ∂u = ( ∂x ∂u ) 2 + ( ∂y ∂u ) 2 + ( ∂z ∂u ) 2 F=∂r ∂u ·∂r ∂v =∂x ∂u ∂x ∂v +∂y ∂u ∂y ∂v +∂z ∂u ∂z ∂v G=∂r ∂ ...
The coordinate curves are The straight-lines, r (θ 0 , z ), 0 ≤z≤h and the circles, r (θ, z 0 ), 0 ≤θ≤ 2 π, where θ 0 and z 0 ...
Example 7-34. Evaluate the integral ∫∫∫ V f(x, y, z )dV, where f(x, y, z) = 6(x+y), dV =dxdydz is an element of volume and V rep ...
Example 7-35. Evaluate the integral ∫∫∫ V F(x, y, z)dV, where F =xˆe 1 +xy ˆe 2 +eˆ 3 and dV =dxdydz is a volume element. The ...
Volume Elements Revisited Consider the volume element dV =dxdydz from cartesian coordinates and intro- duce a change of variable ...
represents a small change in r. One can think of the differential dr as the diagonal of a parallelepiped having the vector sid ...
Cylindrical Coordinates (r,θ,z ) The transformation from rectangular coordinates (x, y, z )to cylindrical coordi- nates (r, θ, z ...
are unit vectors tangent to the coordinate curves, where ˆer·ˆeθ= 0, ˆer · ˆez = 0 and ˆeθ·ˆez= 0. The unit vector ˆez=ˆer׈eθpr ...
Figure 7-27. Coordinate surfaces and coordinate curves for cylindrical coordinates. The partial derivative vectors ∂r ∂ρ = sin ...
The element of volume in spherical coordinates is given by dV =ρ^2 sin θdθdφdρ and the element of surface area is dS =ρ^2 sin θd ...
To evaluate the line integral let x= cos θ, y = sin θwith dx =−sin θ dθ, dy = cos θ dθ and show I 4 = ∫ C F·dr = ∫ C (y−z)dx + ...
Exercises 7-1. Sketch the given surfaces (i) y^2 a^2 + z^2 b^2 = x^2 c^2 (ii) z^2 a^2 + x^2 b^2 = y^2 c^2 , a > b > c 7- ...
7-8. If r =r (t)denotes a space curve, show that the torsion can be calculate from the relation V = r ′·(r ′′ ×r ′′′) (r ...
7-17. Let r (s)denote the position vector of a space curve which is defined in terms of the arc length s. (a) Show that the eq ...
7-21. Find a unit normal vector to the cylinder x^2 +y^2 = 1 7-22. Find a unit normal vector to the sphere x^2 +y^2 +z^2 = 1 ...
7-32. Consider a circle of radius ρ < a centered at x=a > 0 in the xz plane. The parametric equations of this circle are ...
7-37. Evaluate the surface integral I= ∫∫ S f(x, y, z )dS, where f(x, y, z ) = 2(x+ 1)y and Sis the surface of the cylinder x^2 ...
7-41. (Lagrange multipliers) Use Lagrange multipliers to Minimize ω=ω(x, y, z ) = x^2 +y^2 +z^2 , subject to the constraint con ...
7-46. Show that (A×B)·(C×D) = ∣∣ ∣∣A·C A·D B·C B·D ∣∣ ∣∣ 7-47. Let F =F(x, y, z) and G=G(x, y, z ) denote continuo ...
Find the error associated with each data point and then square these errors and sum them to obtain the quantity ∑N i=1 Ei^2 call ...
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