Vectors_Tensors_06_Vector_Calculus_1_Differentiation

Vectors_Tensors_06_Vector_Calculus_1_Differentiation -...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 1.6 Solid Mechanics Part III Kelly 30 1.6 Vector Calculus 1 - Differentiation Calculus involving vectors is discussed in this section, rather intuitively at first and more formally toward the end of this section. 1.6.1 The Ordinary Calculus Consider a scalar-valued function of a scalar , for example the time-dependent density of a material ) ( t ρ = . The calculus of scalar valued functions of scalars is just the ordinary calculus. Some of the important concepts of the ordinary calculus are reviewed in Appendix B to this Chapter, §1.B.2. 1.6.2 Vector-valued Functions of a scalar Consider a vector-valued function of a scalar , for example the time-dependent displacement of a particle ) ( t u u = . In this case, the derivative is defined in the usual way, t t t t dt d t Δ Δ + = Δ ) ( ) ( lim 0 u u u , which turns out to be simply the derivative of the coefficients 1 , i i dt du dt du dt du dt du dt d e e e e u + + = 3 3 2 2 1 1 Partial derivatives can also be defined in the usual way. For example, if u is a function of the coordinates, ) , , ( 3 2 1 x x x u , then 1 3 2 1 3 2 1 1 0 1 ) , , ( ) , , ( lim 1 x x x x x x x x x x Δ Δ + = Δ u u u Differentials of vectors are also defined in the usual way, so that when 3 2 1 , , u u u undergo increments 3 3 2 2 1 1 , , u du u du u du Δ = Δ = Δ = , the differential of u is 3 3 2 2 1 1 e e e u du du du d + + = and the differential and actual increment u Δ approach one another as 0 , , 3 2 1 Δ Δ Δ u u u . 1 assuming that the base vectors do not depend on t
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Section 1.6 Solid Mechanics Part III Kelly 31 Space Curves The derivative of a vector can be interpreted geometrically as shown in Fig. 1.6.1: u Δ is the increment in u consequent upon an increment t Δ in t . As t changes, the end-point of the vector ) ( t u traces out the dotted curve Γ shown – it is clear that as 0 Δ t , u Δ approaches the tangent to Γ , so that dt d / u is tangential to Γ . The unit vector tangent to the curve is denoted by τ : dt d dt d / / u u τ = (1.6.1) Figure 1.6.1: a space curve; (a) the tangent vector, (b) increment in arc length Let s be a measure of the length of the curve Γ , measured from some fixed point on Γ . Let s Δ be the increment in arc-length corresponding to increments in the coordinates, [] T 3 2 1 , , u u u Δ Δ Δ = Δ u , Fig. 1.6.1b. Then, from the ordinary calculus (see Appendix 1.A.2), () ( ) ( ) ( ) 2 3 2 2 2 1 2 du du du ds + + = so that 2 3 2 2 2 1 + + = dt du dt du dt du dt ds But 3 3 2 2 1 1 e e e u dt du dt du dt du dt d + + = so that dt ds dt d = u (1.6.2) ) ( t u ) ( t t Δ + u u Δ τ s Γ 1 x 2 x 1 du 2 du ds s Δ (a) (b)
Background image of page 2
Section 1.6 Solid Mechanics Part III Kelly 32 Thus the unit vector tangent to the curve can be written as ds d dt ds dt d u u τ = = / / ( 1 . 6 . 3 ) If u is interpreted as the position vector of a particle and t is interpreted as time, then dt d / u v = is the velocity vector of the particle as it moves with speed dt ds / along Γ .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.

Page1 / 20

Vectors_Tensors_06_Vector_Calculus_1_Differentiation -...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online