Explanation The gradient of a function is the operation of Del operator on the function. It is a vector quantity. Write the expression for the gradient of a function. ∇ V → = ∂ V ∂ x x ^ + ∂ V ∂ y y ^ + ∂ V ∂ z z ^ …… (1) The divergence of a vector is defined as the measure of spreading the vector from a point. Write the expression for the divergence of a vector. ∇ . V → = ( x ^ ∂ ∂ x + y ^ ∂ ∂ x + z ^ ∂ ∂ x ) . ( V x x ^ + V y y ^ + V z z ^ ) The final expression for divergence reduces as. ∇ . V → = ∂ V x ∂ x + ∂ V y ∂ y + ∂ V z ∂ z …… (2) The curl of a vector is defined as the measure of the quantity of swirl of the vector around a point. Write the expression for the curl of a vector. ∇ × V → = ( x y z ∂ ∂ x ∂ ∂ y ∂ ∂ z V x V y V z ) The final expression for curl reduces as. ∇ × V → = x ^ ( ∂ V z ∂ y − ∂ V y ∂ z ) + y ^ ( ∂ V x ∂ z − ∂ V z ∂ x ) + z ^ ( ∂ V y ∂ x − ∂ V x ∂ y ) ……. (3) Write the expression for the cross product of two vectors. A → × B → = ( x ^ y ^ z ^ A x A y A z B x B y B z ) The final expression for the cross product reduces as. A → × B → = x ^ ( A y B z − B y A z ) + y ^ ( B x A z − A x B z ) + z ^ ( A x B y − B x A y ) …… (4) Write the expression for the gradient of product of two functions. ∇ ( f V → ) = f ∇ V → + V → ∇ f …… (5) Here, f is a scalar function and V is a vector. Write the expression for the product rule of divergence of curl of two vectors. ∇ . ( A → × B → ) = B → . ( ∇ × A → ) − A → . ( ∇ × B → ) …… (6) Here, A → and B → are the two vectors Write the expression for the curl of product of two functions. ∇ × ( f V → ) = f ( ∇ × V → ) + V → × ( ∇ f ) …… (7) Calculation: Substitute ∂ V ∂ x x ^ + ∂ V ∂ y y ^ + ∂ V ∂ z z ^ for ∇ V → and f for V in equation (4). ∇ ( f V → ) = f ( ∂ V ∂ x x ^ + ∂ V ∂ y y ^ + ∂ V ∂ z z ^ ) + V → ( ∂ f ∂ x x ^ + ∂ f ∂ y y ^ + ∂ f ∂ z z ^ ) = ∂ ( f V ) ∂ x x ^ + ∂ ( f V ) ∂ y y ^ + ∂ ( f V ) ∂ z z ^ = ∇ ( f V → ) Substitute A → for V → in equation (3). ∇ × A → = x ^ ( ∂ A z ∂ y − ∂ A y ∂ z ) + y ^ ( ∂ A x ∂ z − ∂ A z ∂ x ) + z ^ ( ∂ A y ∂ x − ∂ A x ∂ y ) Substitute B → for V → in equation (3). ∇ × B → = x ^ ( ∂ B z ∂ y − ∂ B y ∂ z ) + y ^ ( ∂ B x ∂ z − ∂ B z ∂ x ) + z ^ ( ∂ B y ∂ x − ∂ B x ∂ y ) Substitute x ^ ( A y B z − B y A z ) + y ^ ( B x A z − A x B z ) + z ^ ( A x B y − B x A y ) for A → × B → , x ^ ( ∂ A z ∂ y − ∂ A y ∂ z ) + y ^ ( ∂ A x ∂ z − ∂ A z ∂ x ) + z ^ ( ∂ A y ∂ x − ∂ A x ∂ y ) for ∇ × A → and x ^ ( ∂ B z ∂ y − ∂ B y ∂ z ) + y ^ ( ∂ B x ∂ z − ∂ B z ∂ x ) + z ^ ( ∂ B y ∂ x − ∂ B x ∂ y ) for ∇ × B → in equation (6). [ ∇ . ( x ^ ( A y B z − B y A z ) + y ^ ( B x A z − A x B z ) + z ^ ( A x B y − B x A y ) ) ] = [ B → . ( x ^ ( ∂ A z ∂ y − ∂ A y ∂ z ) + y ^ ( ∂ A x ∂ z − ∂ A z ∂ x ) + z ^ ( ∂ A y ∂ x − ∂ A x ∂ y ) ) − A →

4th Edition

ISBN: 9781108420419

Author: David J. Griffiths

Publisher: Cambridge University Press

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Chapter 1.2, Problem 1.21P

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To verify: Product rules for vectors.

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To verify: Product rules for vectors.

Solution Summary: The author explains the product rules for vectors. The gradient of a function is the operation of Del operator on the function.

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