Question

1.) For the scalar potential function φ = (x^2 + y^2 + z^2)^2 and the velocity vector field u=(y^2, z, x^2) calculate the following vector quantities:
a.) ∇φ ; ∇⋅ u
b.) (∇^2)φ = (∇⋅ ∇)φ ; (∇^2)u
c.) ∇ × u
where u is a vector, and the vector operator ∇ = ∂/∂x , ∂/∂y, ∂/∂z)

Answer 1

Answer on Question #45870 – Math – Vector Calculus

Problem.

1.) For the scalar potential function $\phi = (x^{\wedge}2 + y^{\wedge}2 + z^{\wedge}2)^{\wedge}2$ and the velocity vector field $u = (y^{\wedge}2, z, x^{\wedge}2)$ calculate the following vector quantities:

a) $\nabla \phi ;\nabla \cdot u$

b) $(\nabla^{\wedge}2)\phi = (\nabla \cdot \nabla)\phi ;(\nabla^{\wedge}2)u$

c) $\nabla \times u$

where $u$ is a vector, and the vector operator $\nabla = (\partial/\partial x, \partial/\partial y, \partial/\partial z)$

Solution.

By $u_x, u_y, u_z$ we will denote coordinates of vector field $u$.

a) $\nabla \phi = \left(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}\right) = (4x(x^2 +y^2 +z^2),4y(x^2 +y^2 +z^2),4z(x^2 +y^2 +z^2))$ by definition of del operator (or nabla operator).

$\nabla \cdot \vec{u} = \frac{\partial u_x}{\partial x} +\frac{\partial u_y}{\partial y} +\frac{\partial u_z}{\partial z} = \frac{\partial}{\partial x} (y^2) + \frac{\partial}{\partial y} (z) + \frac{\partial}{\partial z} (x^2) = 0 + 0 + 0 = 0$ by definition of inner product of del operator (or nabla operator) and vector field.

b) $(\nabla^2)\phi = \nabla (\nabla \phi) = \nabla \left(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}\right) = \Delta \phi = \frac{\partial^2\phi}{\partial x^2} +\frac{\partial^2\phi}{\partial y^2} +\frac{\partial^2\phi}{\partial z^2} = (4(x^2 +y^2 +z^2) + 8x^2) + (4(x^2 +y^2 +z^2) + 8y^2) = 20(x^2 +y^2 +z^2)$ by definition of scalar Laplacian.

$(\nabla^2)\vec{u} = \left(\nabla^2 u_x,\nabla^2 u_y,\nabla^2 u_z\right) = (2,0,2),$ by definition of vector Laplasian.

c) \nabla \times \vec{u} = \left| \begin{array}{ccc}\hat{\mathbf{x}} & \hat{\mathbf{y}} & \hat{\mathbf{z}}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ u_x & u_y & u_z \end{array} \right| = \left(\frac{\partial u_z}{\partial y} -\frac{\partial u_y}{\partial z},\frac{\partial u_x}{\partial z} -\frac{\partial u_z}{\partial x},\frac{\partial u_y}{\partial x} -\frac{\partial u_z}{\partial y}\right) = \left(\frac{\partial}{\partial y} (x^2) - \frac{\partial}{\partial z} (z),\frac{\partial}{\partial z} (y^2) - \frac{\partial}{\partial x} (x^2),\frac{\partial}{\partial x} (z) - \frac{\partial}{\partial y} (y^2)\right) = (-1, - 2x, - 2y), by definition of vector product of del operator (or nabla operator) and vector field.

Answer:

a) $\nabla \phi = (4x(x^{2} + y^{2} + z^{2}),4y(x^{2} + y^{2} + z^{2}),4z(x^{2} + y^{2} + z^{2})),\nabla \cdot \vec{u} = 0;$

b) $(\nabla^2)\phi = 20(x^2 +y^2 +z^2),(\nabla^2)\vec{u} = (2,0,2)$

c) $\nabla \times \vec{u} = (-1, - 2x, - 2y).$

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