. MA 105: Calculus Lecture 20 . Prof. B.V. Limaye IIT Bombay Friday, 17 March 2017 B.V. Limaye MA 105: Lec-20
. Vector Algebra . We begin a part of this course known as ‘ Vector Analysis ’. It deals with scalar fields and vector fields. We begin with some concepts in ‘ Vector Algebra ’. In Lecture 12, we have defined the Euclidean space R m := { x x x = ( x 1 , . . . , x m ) : x i ∈ R for i = 1 , . . . , m } , where m ∈ N . An element of R 1 := R is called a scalar , and an element of R m is called a vector if m ≥ 2. y := ( y 1 , . . . , y m ) ∈ R m , and a ∈ R . We Let x x x := ( x 1 , . . . , x m ) , y y have already defined the sum x x x + y y y and the scalar multiple ax x x . Also, we have studied the scalar product of x x x and y y y : x · y x y y := x 1 y 1 + · · · + x m y m ∈ R . x It is often called the dot product of x x and y y . x y B.V. Limaye MA 105: Lec-20
. Vector Product . Let m := 3, x x x := ( x 1 , x 2 , x 3 ) and y y y := ( y 1 , y 2 , y 3 ). We now define the vector product of x x x with y y y : y := ( x 2 y 3 − x 3 y 2 , x 3 y 1 − x 1 y 3 , x 1 y 2 − x 2 y 1 ) ∈ R 3 . x x x × y y It is often called the cross product of x x x with y y y . Let i i i := (1 , 0 , 0) , j j j := (0 , 1 , 0) , k k k := (0 , 0 , 1) . Then x x := x 1 i x i i + x 2 j j j + x 3 k k k , y y y := y 1 i i i + y 2 j j j + y 3 k k k , and � � � � i i i j j j k k k � � � � x x × y y = x 1 x 2 x 3 . x y � � � � y 1 y 2 y 3 Note: i i i × j j j = k k k , j j j × k k k = i i , k i k k × i i i = j j j . B.V. Limaye MA 105: Lec-20
z ∈ R 3 , Properties of a determinant show that for every z z ( x x x + y y y ) × z z z = ( x x × z x z z ) + ( y y y × z z z ) and y × x y y x x = − ( x x x × y y y ) . Let x x x ̸ = 0 0 0 and y y y ̸ = 0 0 0. It can be shown that x x x × y y y = ∥ x x x ∥ ∥ y y y ∥ (sin θ ) n n n , where θ ∈ [0 , π ] is the angle between x x x and y y y , and n n n is the unit vector which is perpendicular to the plane containing x x x and y y y , and obeys the ‘right-hand rule’. x × y y n θ x Hence ∥ x x × y x y y ∥ = the area of the parallelogram with sides x x x , y y y . B.V. Limaye MA 105: Lec-20
. Scalar Triple Product and Vector triple Product . Let x x , y y , z z ∈ R 3 . Then x x · ( y y × z z ) ∈ R and x x × ( y y × z z ) ∈ R 3 x y z x y z x y z are called the scalar triple product and the vector triple product of x x x , y y y , z z z respectively. It is easy to see that if x x x = ( x 1 , x 2 , x 3 ) , y y y = ( y 1 , y 2 , y 3 ) , z z z = ( z 1 , z 2 , z 3 ) , then � � � � x 1 x 2 x 3 � � � � x x · ( y y y × z z z ) = . x y 1 y 2 y 3 � � � � z 1 z 2 z 3 Geometrically, x x x · ( y y y × z z ) can be interpreted as the (signed) z volume of the parallelopiped defined by the vectors x x , y y , z z . x y z One can prove the Lagrange formula x x x × ( y y y × z z ) = ( x z x x · z z z ) y y y − ( x x x · y y y ) z z z by considering each component of the LHS and the RHS. B.V. Limaye MA 105: Lec-20
. Scalar Fields and Vector Fields . A scalar field is an assignment of a scalar to each point in a region in the space. For example, the temperature at a point on the earth is a scalar field (defined on a subset of R 3 .) A vector field is an assignment of a vector to each point in a region in the space. For example, the velocity field of a moving fluid is a vector field that associates a velocity vector to each point in the fluid. . Definition . Let m ∈ N , and let D be a subset of R m . A scalar field is a map from D to R . A vector field is a map from D to R m . If m = 2, then it is called a vector field in the plane , and if m = 3, then it is called a vector field in the space . . B.V. Limaye MA 105: Lec-20
. Smooth Scalar and Vector Fields . Suppose D is an open subset of R m , that is, every point in D is an interior point of D . A scalar field f : D → R is called smooth if all first order partial derivatives of f exist and are continuous on D . The set of all smooth scalar fields on D is denoted by C 1 ( D ). Similarly, the set of all scalar fields on D having continuous partial derivatives of the first and second order is denoted by C 2 ( D ). F : D → R m be a vector field on D , and let Let F F F ( x x ) = ( F 1 ( x x ) , . . . , F m ( x x )) for x x ∈ D , F F x x x x where F i : D → R is scalar field on D , i = 1 , . . . , m . Then the vector field F F F is called smooth if each component scalar field F i is smooth, that is, if each ∂ F i , i , j = 1 , . . . , m exists and is ∂ x j continuous on D . B.V. Limaye MA 105: Lec-20
. Gradient, Divergence and Curl . Let f be a smooth scalar field on D ⊂ R 3 . Then the vector field ( ∂ f ) ∂ x , ∂ f ∂ y , ∂ f grad f := ∇ f = ∂ z defined on D is called the gradient field of f . F := ( P , Q , R ) be a smooth vector field on D ⊂ R 3 . Next, let F F The divergence field of F F F is the scalar field on D defined by F = ∂ P ∂ x + ∂ Q ∂ y + ∂ R div F F := ∇ · F F F ∂ z , and the curl field of F F F is the vector field on D defined by ( ∂ R ) ∂ y − ∂ Q ∂ z , ∂ P ∂ z − ∂ R ∂ x , ∂ Q ∂ x − ∂ P curl F F := ∇ × F F F F = . ∂ y B.V. Limaye MA 105: Lec-20
As in the case of the cross product x x × y x y y , we may write � � � � i i i j j j k k k � � � � ∂ ∂ ∂ � � ∇ × F F = ∇ × ( P , Q , R ) F = . � � ∂ x ∂ y ∂ z � � � � P Q R For any m ∈ N , we can define the gradient field grad f of a scalar field f on an open subset of R m as well as the divergence field div F F F of a vector field F F F on an open subset of R m in a similar manner. But the curl field curl F F F can only be F on an open subset of R 3 . defined for a vector field F F Of course, if D is an open subset of R and f is a smooth scalar field on D , then we can let F F F ( x , y , z ) := ( f ( x ) , 0 , 0) for ( x , y , z ) ∈ D × R 2 , and define curl f := curl F F F . Also, if D is an open subset of R 2 and G G G is a smooth vector field on D , then we can let F F F ( x , y , z ) := ( G ( x , y ) , 0) for ( x , y , z ) ∈ D × R , and define curl G G G := curl F F F . B.V. Limaye MA 105: Lec-20
. The GCD Sequence . Suppose the first and second order partial derivatives of f and of P , Q , R exist and are continuous on D . By the Mixed Partials Theorem, we obtain (i) curl (grad f ) = ∇ × ( ∇ f ) = 0 0 0: ( ∂ 2 f ) ∂ y ∂ z − ∂ 2 f ∂ z ∂ y , ∂ 2 f ∂ z ∂ x − ∂ 2 f ∂ x ∂ y − ∂ 2 f ∂ 2 f ∂ x ∂ z , = (0 , 0 , 0) , ∂ y ∂ x and also (ii) div (curl F F ) = ∇ · ( ∇ × F F F F ) = 0: ( ∂ R ) ( ∂ P ) ( ∂ Q ) ∂ ∂ y − ∂ Q + ∂ ∂ z − ∂ R + ∂ ∂ x − ∂ P = 0 . ∂ x ∂ z ∂ y ∂ x ∂ z ∂ y Thus Gradient, Curl and Divergence give a sequence of maps { scalar } { vector } { vector } { scalar } grad curl div − → − → − → fields fields fields fields which satisfies curl (grad f ) = 0 0 0 and div (curl F F F ) = 0, so that the successive composites are zero. B.V. Limaye MA 105: Lec-20
The above phenomenon raises the following basic questions: 0 (i) If G G G is a smooth vector field such that curl G G G = 0 0, then is G a gradient field, that is, is there a scalar field f such that G G G G G = grad f ? (ii) If H H H is smooth vector field such that div H H H = 0, then is H H H a curl field, that is, is there a vector field F F F such that H H H = curl F F F ? These questions are reminiscent of the Fundamental theorem of Calculus, Part I, which answers the following question in the affirmative: If g is a continuous function on [ a , b ], then is there an antiderivative of g , that is, is there a differentiable function f such that g = f ′ ? As such, the two questions raised above call for a suitable theory of integration, to which we now turn. Eventually, we shall come back to these questions. B.V. Limaye MA 105: Lec-20
. Laplacian . Let f be a smooth vector field on D ⊂ R 3 , and suppose the second order partials f xx , f yy , f zz exist on D . Let us consider the maps { scalar } { vector } { scalar } grad div − → − → . fields fields fields The Laplacian field of f is the scalar field defined on D by div (grad f ) := ∇ · ( ∇ f ) = ∇ 2 f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ 2 y + ∂ 2 f ∂ 2 z . The Laplacian plays a very important role in the theory of partial differential equations, and its various applications. B.V. Limaye MA 105: Lec-20
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