Problem involving del operator in vector analysis

= \frac{1}{g} \left(\frac{\partial f}{\partial x}i +\frac{\partial f}{\partial y}j + \frac{\partial f}{\partial z}k\right) - \frac{f}{g^2}\left(\frac{\partial g}{\partial x}i + \frac{\partial g}{\partial y}j + \frac{\partial g}{\partial z}k\right)= \frac{1}{g} \nabla f - \frac{f}{g^2} \nabla g= \frac{1}{g} \left(\nabla f - \frac{f}{g}
  • #1
Sudip Maity
1
0

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.
 
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  • #2
Use the definition of the del operator; otherwise you are just staring at upside-down triangles wondering what to do.

[tex]\nabla (f/g) = \frac{\partial (f/g)}{\partial x}i +\frac{\partial (f/g)}{\partial y}j + \frac{\partial(f/g)}{\partial z}k[/tex]
 
  • #3


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)∂
 

FAQ: Problem involving del operator in vector analysis

1. What is the del operator in vector analysis?

The del operator, denoted as ∇ (nabla), is a mathematical operator used in vector analysis to represent the gradient, divergence, and curl of a vector field.

2. How is the del operator used to calculate the gradient of a vector field?

The gradient of a vector field is calculated using the del operator as follows: ∇f = (df/dx, df/dy, df/dz), where f is the scalar field and dx, dy, dz are the partial derivatives in the x, y, and z directions, respectively.

3. What is the physical meaning of the divergence of a vector field?

The divergence of a vector field, represented by ∇ · F, measures the rate at which the vector field is flowing out of a given point. A positive divergence indicates that the vector field is spreading out, while a negative divergence indicates that the vector field is converging.

4. Can the del operator be applied to any type of vector field?

Yes, the del operator can be applied to any type of vector field, as long as the vector field is differentiable. It is commonly used in electromagnetism, fluid dynamics, and other fields of physics and engineering.

5. How is the curl of a vector field calculated using the del operator?

The curl of a vector field is calculated using the del operator as follows: ∇ x F = (dFz/dy - dFy/dz, dFx/dz - dFz/dx, dFy/dx - dFx/dy), where F is the vector field and dx, dy, dz are the partial derivatives in the x, y, and z directions, respectively.

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