Problem involving del operator in vector analysis

[Sudip Maity]

Homework Statement

prove grad(f/g)=((g grad f)-(f grad g))/g^2,if g not equal to 0.

Homework Equations

no idea.

The Attempt at a Solution

rhs will be grad f -(f grad g)/g^2.
can't make out what 2 do after that.referred other books but no help.it's a very obscure identity.found this in m.r.spiegel's vector analysis no.57 pg-78,chap GDC.

 

[Phrak]

Use the definition of the del operator; otherwise you are just staring at upside-down triangles wondering what to do.

$$\nabla (f/g) = \frac{\partial (f/g)}{\partial x}i +\frac{\partial (f/g)}{\partial y}j + \frac{\partial(f/g)}{\partial z}k$$

 

[kerimek]

I can provide some guidance on how to approach this problem. First, let's review the del operator in vector analysis. The del operator, represented by the symbol ∇, is a vector operator that is used to calculate derivatives in three-dimensional space. It is defined as:

∇ = i∂/∂x + j∂/∂y + k∂/∂z

where i, j, and k are unit vectors in the x, y, and z directions, respectively.

Now, let's look at the given problem. We are asked to prove that grad(f/g) = ((g grad f) - (f grad g))/g^2, if g is not equal to 0. To do this, we need to use the properties of the del operator and the rules of vector calculus.

First, let's rewrite the left-hand side of the equation using the definition of the gradient:

grad(f/g) = (∇(f/g)) = (∂/∂x + ∂/∂y + ∂/∂z)(f/g)

Next, we can use the product rule for derivatives to expand this expression:

= (∂f/∂x/g + f∂/∂x(1/g) + ∂f/∂y/g + f∂/∂y(1/g) + ∂f/∂z/g + f∂/∂z(1/g))

= (∂f/∂x/g + f∂/∂x(1/g) + ∂f/∂y/g + f∂/∂y(1/g) + ∂f/∂z/g + f∂/∂z(1/g))

= (∂f/∂x/g + f∂/∂x(1/g) + ∂f/∂y/g + f∂/∂y(1/g) + ∂f/∂z/g + f∂/∂z(1/g))

Now, we can use the quotient rule to simplify the term f∂/∂x(1/g):

= (∂f/∂x/g + f(-1/g^2)∂g/∂x + ∂f/∂y/g + f(-1/g^2)∂