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vector calculus qs: dal and scalars

hey
i have a question on one of my worksheets which i have solved mostly, but have ended up with one term given as:

x^2 Del

Del being the vector differential operator (Nabla), and x is a scalar.

I need to get in a form where the x term will be after the Del. can these be switched around (i.e. Del x^2), or do i have to do something else to them, such as differentiating the x^2?? hope this makes sense!
Original post by EmergencyBagels
hey
i have a question on one of my worksheets which i have solved mostly, but have ended up with one term given as:

x^2 Del

Del being the vector differential operator (Nabla), and x is a scalar.

I need to get in a form where the x term will be after the Del. can these be switched around (i.e. Del x^2), or do i have to do something else to them, such as differentiating the x^2?? hope this makes sense!


No you cannot do this. Note that ∇=(∂∂x∂∂y∂∂z)\nabla = \begin{pmatrix} \dfrac{\partial}{\partial x} \\[0.3cm] \dfrac{\partial }{\partial y} \\[0.3cm] \dfrac{\partial}{\partial z} \end{pmatrix} is an operator. You apply this onto some scalar/vector field by putting it after it, related by either a dot or cross product.

Here you have that
x2∇=(x2∂∂xx2∂∂yx2∂∂z)x^2 \nabla = \begin{pmatrix} x^2 \dfrac{\partial}{\partial x} \\[0.3cm] x^2 \dfrac{\partial }{\partial y} \\[0.3cm] x^2 \dfrac{\partial}{\partial z} \end{pmatrix}

is again an operator, just a slightly different one. You cannot write it as

∇x2=(∂∂x(x2)∂∂y(x2)∂∂z(x2))=(2x00)\nabla x^2 = \begin{pmatrix} \dfrac{\partial}{\partial x}(x^2) \\[0.3cm] \dfrac{\partial }{\partial y}(x^2) \\[0.3cm] \dfrac{\partial}{\partial z}(x^2) \end{pmatrix} = \begin{pmatrix} 2x \\ 0 \\ 0 \end{pmatrix}

as this, on the other hand, is a vector. You do not apply it to something like you would with an operator.

If you need help with a question, then post it on the maths forum (https://www.thestudentroom.co.uk/forumdisplay.php?f=38) with your working out and we can help you from there when it comes to dealing with your confusion here.
(edited 5 years ago)

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