Equation for finding the gradient in spherical coordinates

[KUphysstudent]

<Mentor note: moved from a technical forum and therefore without template>So I´m trying to understand how to use the equation for finding the gradient in spherical coordinates, just going from cartesian to spherical seemed crazy. Now I´m at a point where I want to try out what I have read and I immediately run into problems, which clearly tells me I have no idea what I´m doing.

Problem I was trying to solve:
Given a scalarfield β = A/r where r = (x^2+y^2+z^2)^1/2 and A is a konstant, calculate the gradient in spherical coordinates.

∇β = ∂β/∂r i_r + 1/r ∂β/∂θ i_θ + 1/rsinθ ∂β/∂φ i_φ

When I thought the solution was pretty simply and then I go to the back of my book to check the result and I´m not even close.
How on Earth does the result become negative? it is also negative in Cartesian coordinates which don´t understand either.
Well that is basically my frustration, how does this become negative?

 

[BvU]

Well that is basically my frustration, how does this become negative

If something becomes smaller, the derivative is negative !
From your $\nabla\beta$ in spherical coordinates, all that remains is the $\partial \over \partial r$ and the exponent of $r$ in $\beta$ is -1