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Vector Differentiation - Vector Differentiation 1 Ordinary...

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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 ...
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