Calculating the Gradient of Magnitude of a Scalar Field

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In summary, the speaker tried to calculate the gradient of the magnitude of the gradient of a scalar field and arrived at an integral. They then made substitutions and treated a certain term as constant, leading to a simplified form. However, they are unsure if their process is correct and are asking for clarification.
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ShayanJ
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I know my question is strange(and maybe stupid) but I'm really curious about it.I once tried to calculate the gradient of the magnitude of the gradient of a scalar field which for its x coordinate,I found:

[itex]
\left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =
\frac {\frac { \partial ^ {2} \varphi } {\partial x ^ {2} } }
{\sqrt { 1+ (\frac{\frac {\partial \varphi} {\partial y} } {\frac {\partial \varphi } {\partial x} } )^{2} } }

[/itex]

then I tried to write it in a simpler form and it just came into my mind that I can integrate it and write it as the derivative of a function so I made substitutions below and treated [itex] \frac {\partial \varphi} {\partial y} [/itex] as constant:

[itex]\frac{\frac {\partial \varphi } {\partial y} }{\frac {\partial \varphi } {\partial x} }=\tan{\theta}
[/itex]
[itex]
\frac{\partial ^{2} \varphi }{\partial x ^{2} }=- \frac {\partial \varphi } {\partial y} \csc ^{2}{\theta} d \theta
[/itex]

so I arrived at:

[itex]

\int \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =
\frac {\partial \varphi} {\partial y} \csc {\theta}

[/itex]

I reverted and got the following:

[itex]

\int \left[ \nabla \left( \left| \nabla \varphi \right| \right) \right]_{x} =
\left| \nabla \varphi \right|

[/itex]

Now comes my question:
When I check the things I've done,I just can tell bullsh*t.But the result tells that was really an inegration but I really don't understand how can that be an integration.

Sorry for such a mess and thanks in advance
 
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  • #2
I'm not really sure what you did (I'm not really good at that stuff), but I don't think you have to treat [itex]\frac {\partial \varphi} {\partial y}[/itex] as constant since [itex]\theta[/itex] can be a variable (I'm not really sure if I'm right, but I think it works that way).
 

1. What is the gradient of magnitude of a scalar field?

The gradient of magnitude of a scalar field is a vector quantity that represents the direction and magnitude of the steepest increase in the scalar field at a given point.

2. How do you calculate the gradient of magnitude of a scalar field?

The gradient of magnitude of a scalar field can be calculated by taking the partial derivatives of the scalar field with respect to each variable, and then combining these derivatives into a vector using the gradient operator (∇).

3. What is the physical significance of the gradient of magnitude of a scalar field?

The gradient of magnitude of a scalar field is important in physics and engineering as it can be used to determine the direction and rate of change of a scalar quantity, such as temperature or pressure, in a given region.

4. Can the gradient of magnitude of a scalar field ever be zero?

Yes, the gradient of magnitude of a scalar field can be zero at points where the scalar field is constant, meaning there is no change in the scalar quantity in any direction.

5. How does the gradient of magnitude of a scalar field relate to the concept of level sets?

The gradient of magnitude of a scalar field is perpendicular to the level sets of the scalar field, meaning it points in the direction of the steepest increase in the scalar quantity. This relationship is often used in contour maps to represent changes in elevation or other scalar values.

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