Question

i am struggling with this question can anyone help please?

Calculate the divergence of the following vector field ~v(x, y, z) and
evaluate it at the point P as indicated.
v(x, y, z) = (e^x , ln(xy) , e^(xyz)) = e^(x)i + ln(xy)j + e^(xyz)k,
P = (2, 1,−1),
where ln is the natural logarithm.
i, v, k and j are vectors

what is div v
and div v (2,1,-1)

thank you x

Answer 1

So, the del operator is defined as
\[\vec{\nabla} = <\frac{\partial}{\partial x},\frac{\partial}{\partial y} , \frac{\partial}{\partial z}>\]
and the divergence of a vector field is the dot product
\[\vec{\nabla} \centerdot \vec{F}\]

Answer 2

grad Phi=<δ Phiδ x,δ Phiδ yδ Phiδ z> So take the derivative of e^x wrt x, ln(xy) wrt y, and e^(xyz) wrt to z. I assume you know what the derivative is. Evaluating at those points is trivial.

Answer 3

So if
\[\vec{v}(x,y,z) = <e^x,\ln(xy),e^{xyz}> \]
then
\[\vec{\nabla} \centerdot \vec{v} = \frac{\partial}{\partial x} e^x + \frac{\partial}{\partial y} \ln(xy) + \frac{\partial}{\partial z}e^{xyz} = e^x + \frac{1}{y} + (xy)e^{xyz}\]

Answer 4

I should also point out that the gradient operator acts on a scalar and not a vector, so the above post may be somewhat confusing. There are three common operations using the del operator, and they're called gradient, divergence, and curl. Gradients act on scalar functions, divergences and curls on vector functions, and they are defined as
\[grad(f) = \vec{\nabla}f = <\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}> \]
\[div(\vec{f}) = \vec{\nabla} \centerdot \vec{f}\]
\[curl(\vec{f}) = \vec{\nabla} \times \vec{f}\]