Chapter 3, Problem 26P

(a)

To determine

To prove the expression: $\nabla \left(UV\right)=U\nabla V+V\nabla U$.

Explanation of Solution

Given:

The two scalar functions $U$ and $V$.

Calculation:

The gradient of the function $\left(UV\right)$ is $\nabla \left(UV\right)$.

Calculate the gradient of the function $\nabla \left(UV\right)$ using the relation.

  $\nabla \left(UV\right)=\frac{\partial \left(UV\right)}{\partial x}{a}_x+\frac{\partial \left(UV\right)}{\partial y}{a}_y+\frac{\partial \left(UV\right)}{\partial z}{a}_z$

  $\begin{array}{l}\nabla \left(UV\right)=U\frac{\partial \left(V\right)}{\partial x}{a}_x+V\frac{\partial \left(U\right)}{\partial x}{a}_x+U\frac{\partial \left(V\right)}{\partial y}{a}_y+V\frac{\partial \left(U\right)}{\partial y}{a}_y+U\frac{\partial \left(V\right)}{\partial z}{a}_z+V\frac{\partial \left(U\right)}{\partial z}{a}_z\\ \nabla \left(UV\right)=U\frac{\partial \left(V\right)}{\partial x}{a}_x+U\frac{\partial \left(V\right)}{\partial y}{a}_y+U\frac{\partial \left(V\right)}{\partial z}{a}_z+V\frac{\partial \left(U\right)}{\partial x}{a}_x+V\frac{\partial \left(U\right)}{\partial y}{a}_y+V\frac{\partial \left(U\right)}{\partial z}{a}_z\\ \nabla \left(UV\right)=U\left(\frac{\partial \left(V\right)}{\partial x}{a}_x+\frac{\partial \left(V\right)}{\partial y}{a}_y+\frac{\partial \left(V\right)}{\partial z}{a}_z\right)+V\left(\frac{\partial \left(U\right)}{\partial x}{a}_x+\frac{\partial \left(U\right)}{\partial y}{a}_y+\frac{\partial \left(U\right)}{\partial z}{a}_z\right)\\ \nabla \left(UV\right)=U\nabla \left(V\right)+V\nabla \left(U\right)\end{array}$

The expression: $\nabla \left(UV\right)=U\nabla \left(V\right)+V\nabla \left(U\right)$ is proved.

(b)

To determine

To verify the part (a) assuming $V=5x^{2}y+2yz$ and $U=3xyz$.

Explanation of Solution

Given:

The function $V$ is $5x^{2}y+2yz$.

The function $U$ is $3xyz$.

Calculation:

Calculate the function $\left(UV\right)$ using the relation.

  $\begin{array}{l}UV=\left(3xyz\right)\left(5x^{2}y+2yz\right)\\ UV=15x^3{y}^{2}z+6x{y}^2{z}^2\end{array}$

Calculate the gradient of the function $\left(\nabla \left(UV\right)\right)$ using the relation.

  $\begin{array}{l}\nabla \left(UV\right)=\frac{\partial \left(15x^3{y}^{2}z+6x{y}^2{z}^2\right)}{\partial x}{a}_x+\frac{\partial \left(15x^3{y}^{2}z+6x{y}^2{z}^2\right)}{\partial y}{a}_y+\frac{\partial \left(15x^3{y}^{2}z+6x{y}^2{z}^2\right)}{\partial z}{a}_z\\ \nabla \left(UV\right)=\left(45x^2{y}^{2}z+6y^2{z}^2\right)a_x+\left(30x^{3}yz+12xy{z}^2\right)a_y+\left(15x^3{y}^2+12x{y}^{2}z\right)a_z\end{array}$

Calculate the gradient of the function $U$ $\left(\nabla \left(U\right)\right)$ using the relation.

  $\nabla \left(U\right)=\frac{\partial \left(U\right)}{\partial x}{a}_x+\frac{\partial \left(U\right)}{\partial y}{a}_y+\frac{\partial \left(U\right)}{\partial z}{a}_z$

  $\begin{array}{l}\nabla \left(U\right)=\frac{\partial \left(3xyz\right)}{\partial x}{a}_x+\frac{\partial \left(3xyz\right)}{\partial y}{a}_y+\frac{\partial \left(3xyz\right)}{\partial z}{a}_z\\ \nabla \left(U\right)=\left(3yz\right)a_x+\left(3xz\right)a_y+\left(3xy\right)a_z\end{array}$

Calculate the gradient of the function $V$ $\left(\nabla \left(V\right)\right)$ using the relation.

  $\nabla \left(V\right)=\frac{\partial \left(V\right)}{\partial x}{a}_x+\frac{\partial \left(V\right)}{\partial y}{a}_y+\frac{\partial \left(V\right)}{\partial z}{a}_z$

  $\begin{array}{l}\nabla \left(V\right)=\frac{\partial \left(5x^{2}y+2yz\right)}{\partial x}{a}_x+\frac{\partial \left(5x^{2}y+2yz\right)}{\partial y}{a}_y+\frac{\partial \left(5x^{2}y+2yz\right)}{\partial z}{a}_z\\ \nabla \left(V\right)=\left(10xy\right)a_x+\left(5x^2+2z\right)a_y+\left(2y\right)a_z\end{array}$

Calculate the sum $\left(U\nabla \left(V\right)+V\nabla \left(U\right)\right)$ using the relation.

  $\begin{array}{l}U\nabla \left(V\right)+V\nabla \left(U\right)=\left(3xyz\right)\left[\begin{array}{l}\left(10xy\right)a_x\\ +\left(5x^2+2z\right)a_y+\left(2y\right)a_z\end{array}\right]+\left(5x^{2}y+2yz\right)\left[\begin{array}{l}\left(3yz\right)a_x+\left(3xz\right)a_y\\ +\left(3xy\right)a_z\end{array}\right]\\ U\nabla \left(V\right)+V\nabla \left(U\right)=\left[\begin{array}{l}\left(30x^2{y}^{2}z\right)a_x+\left(15x^{3}yz+6xy{z}^2\right)a_y+\left(6x{y}^{2}z\right)a_z+\\ \left(15x^2{y}^{2}z+6y^{2}z\right)a_x+\left(15x^{3}yz+6xy{z}^2\right)a_y+\left(15x^3{y}^2+6x{y}^{2}z\right)\end{array}\right]\\ U\nabla \left(V\right)+V\nabla \left(U\right)=\left(45x^2{y}^{2}z+6y^2{z}^2\right)a_x+\left(30x^{3}yz+12xy{z}^2\right)a_y+\left(15x^3{y}^2+12x{y}^{2}z\right)a_z\end{array}$

Thus, the expression: $\nabla \left(UV\right)=U\nabla V+V\nabla U$ is verified.