.Scalar and Vector Function

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Introduction: Vector is a quantity having both magnitude and direction Eg: Force, Velocity.Scalar and Vector Function

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10MAT11 UNIT -4VECTOR CALCULUSIntroduction: Vector is a quantity having both magnitude and direction.Eg: Force, Velocity.Scalar and Vector Function:Scalar point Function: If to every point (x,y,z) of a region R in space there corresponds a scalar then is called a scalar point function and we say that a scalar field is defined in R.Eg: 1. 2. Vector point Function: If to every point (x,y,z) of a region R in space there corresponds a vector then is called a vector point function and we say that a vector field is defined in R.Eg:1. 2. Vector function of single variable: Let the position vector of a point P(x,y,z) in space be If x,y,z are all functions of a single parameter t, then is said to be a vector function of t and also known as Vector point Function and denoted by Hence is called vector equation of the curve.Now, is a vector along the tangent to the curve at P.If t is the time variable then is the velocity of the particle at time t.Also, is the rate of change of velocity i.e., it gives acceleration.Some observations:Operators:i) The vector differential operator read as Nabla or Del is defined by ii) The Laplacian operator is defined by =Gradient, Divergence,Curl and LaplacianGradient of a scalar field: If is continuously differentiable scalar function then the gradient of or grad is defined by i.e.,grad = = Note: grad is a scalar quantity.Geometrical meaning of the gradient:If is a scalar function then gr

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