- #1

ConorDMK

- 25

- 0

## Homework Statement

Let T(

__r__) be a scalar field. Show that, in spherical coordinates

__∇__T = (∂T/∂r) rˆ + (1/r)(∂T/∂θ) θˆ + (1/(r*sin(θ)))(∂T/∂φ) φˆ

Hint. Compute T(

__r__+d

__l__)−T(

__r__) = T(r+dr, θ+dθ, φ+dφ)−T(r, θ, φ) in two different ways for the infinitesimal displacement d

__l__= dr rˆ + rdθ θˆ + r*sin(θ)dφ φˆ and compare the two results.

## Homework Equations

__∇__= (∂/∂x)xˆ + (∂/∂y)yˆ + (∂/∂z)zˆ

## The Attempt at a Solution

dT(

__r__) ≡ T(

__r__+d

__l__)-T(

__r__) = T(r+dr, θ+dθ, φ+dφ) - T(r,θ,φ) = (T(r,θ,φ) + (∂T(

__r__)/∂r)dr + (∂T(

__r__)/∂θ)dθ + (∂T(

__r__)/∂φ)dφ) - T(r,θ,φ)

⇒ dT(

__r__) = (∂T(

__r__)/∂r)dr + (∂T(

__r__)/∂θ)dθ + (∂T(

__r__)/∂φ)dφ

But I don't know where I can go from here, and I don't think what I've done previously is correct (I rubbed out some of the work that continued form this, as I don't know what I can and can't use.)