Vector Differentiation - Vector Differentiation 1. Ordinary...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Vector Differentiation 1. Ordinary Derivatives of Vectors Let be a vector depending on a single variable u. = The ordinary derivative of with respect to the scalar u is given by is a vector depends on u. Therefore the derivative of exists and it is denoted by . Similarly third and higher order derivatives can be obtained. 2 Geometric Representation of Vector Derivative Let be the position vector of a point P(x,y,z) w.r.t. the origin O.Then As u changes P describes a space curve having the parameter u. is a vector in the direction of When If exists, represents a vector in the direction of the tangent to the space curve C at P(x,y,z). If is a unit vector in the direction of the tangent to C at P then diagram Example : Find and to the curve traced by at t. 1 3 Properties of Derivatives If are differentiable vector functions of a scalar u and of u . 1. 2. 3. 4. 5. 6. = where ( is a differentiable scalar function 4 Partial Derivatives of vectors If is a vector function of x,y,z [i.e. the partial derivatives of w.r.t. x,y,z are given by As in calculus higher partial derivatives are defined as etc... Remark : If has continuous partial derivatives, order of differentiation is no matter. Properties of Partial Derivatives : 1 2 3 = 2 5 Differentials of Vectors 1 If 2 4 4 If then then 6 Applications of Vector Differentiation 6.1 Differential Geometry (This is a study of space curves and surfaces.) If s is the arc length from A to B traced by the position vector Proof : Let P(x,y,z) and Q(x+x,y+y,z+z) be two adjacent points on C. *diagram then = + + When If C is a space curve defined by the function from a fixed point on C then where the scalar s is the arc length measured is a unit tangent vector to C and is denoted by is a unit vector in the direction at a given point on C is normal to the curve at that point. If of the normal ; . 3 k is called the curvature of C at that point. (since k > 0) Radius of curvature . 6.2 Mechanics (a) If is the position vector of a moving point (b) If then is the force acting on an object of mass m moving with velocity = ) Self Assessment Questions 1 If 2 If (a) (b) and (c) find , , . find 3 If has constant magnitude show that and are perpendicular. 4 Find the unit tangent vector to the curve x = t2 + 1, y = 4t 3, z = 2t2 6t at the point where t = 2. 5 A particle moves along the curve x = 2t2 , y = t2 4t , z = 3t 5 where t is the time . Find the components of its velocity and acceleration at t = 1. 6 The position vector of a moving particle is given by Show that (a) (b) (c) is perpendicular to is directed toward the origin. where is a constant. 4 ...
View Full Document

This note was uploaded on 10/30/2011 for the course MECHATRONI EN11ME2050 taught by Professor Perera during the Spring '11 term at Asian Institute of Technology.

Ask a homework question - tutors are online