How do you show the following are true for the two vectors 𝑢̅ = 4𝑥𝑦𝑖̅ − 2𝑧𝑗̅ + 11𝑥3𝑘̅ and 𝑣 ̅ = 𝑥 𝑦 𝑖 ̅ + 3 𝑥 2 𝑗 ̅ + 6 𝑘̅ 𝑧? a. ∇×(𝑢̅+𝑣̅)=∇×𝑢̅+∇×𝑣̅ b. ∇.(𝑢̅+𝑣̅)=∇.𝑢̅+∇.𝑣̅ Thank you!

1 Answer
Ultrilliam
May 26, 2018

See below

Explanation:

It is actually easier to show this in the generality, ie that curl is a linear operator :

  • # implies bb nabla times (bb u + bb v)= bb nabla times bb u + bb nabla times bb v#

Start with:

# bb nabla times (bb u + bb v) = det ((bb hat x, bb hat y, bb hat z),(partial x, partial y, partial z),(u_x+ v_x, u_y+ v_y, u_z+ v_z))#

#= bb hat x (partial_y (u_z + v_z) - partial_z (u_y + v_y)) - bb hat y (partial_x(u_z + v_z) - partial_z (u_x + v_x))) + bb hat z (partial_x(u_y + v_y) - partial_y (u_x + v_x))) #

#= ( (partial_y (u_z + v_z) - partial_z (u_y + v_y)), (partial_x(u_z + v_z) - partial_z (u_x + v_x)), (partial_x(u_y + v_y) - partial_y (u_x + v_x)) ) #

Because the differential operator is linear:

#= ( (partial_y (u_z) - partial_z (u_y )), (partial_x(u_z ) - partial_z (u_x )), (partial_x(u_y ) - partial_y (u_x )) ) + ( (partial_y ( v_z) - partial_z ( v_y)), (partial_x( v_z) - partial_z ( v_x)), (partial_x( v_y) - partial_y ( v_x)) ) #

#= det ((bb hat x, bb hat y, bb hat z),(partial x, partial y, partial z),(u_x , u_y , u_z )) + det ((bb hat x, bb hat y, bb hat z),(partial x, partial y, partial z),(u v_x, v_y, v_z))#

#= bb nabla times bb u + bb nabla times bb v#

Similarly:

#bb nabla * ( bb u + bb v) #

#= partial_x ( u_ x + v_ x) + partial_y ( u_ y + v_ y) + partial_z ( u_ z + v_ z)#

Because the differential operator is linear:

# = ( partial_x u_ x + partial_y u_ y + partial_z u_ z ) + ( partial_x v_ x + partial_y v_ y + partial_z v_ z ) #

#= bb nabla * bb u + bb nabla * bb v#

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