Problem 24 - A Course in Mathematical Methods for Physicists

Question

Let \(C\) be a closed curve and \(D\) the enclosed region. Prove the identity $$ \int_{C} \phi \nabla \phi \cdot \mathbf{n} d s=\int_{D}\left(\phi \nabla^{2} \phi+\nabla \phi \cdot \nabla \phi\right) d A. $$

Step-by-Step Solution

Answer

By applying divergence theorem to the line integral, transforming it into a double integral, calculating the divergence of the functional vector, and showing that the results obtained are the same, we can conclude that the given identity is proven.

Step 1: Preparations

Write down the given expression and all the known values and formulas that will be useful. Most importantly remember the divergence theorem, \(\int_{C} F \cdot n ds = \int_{V} \nabla \cdot F dA\). The functional vector \(F\) in this case is \(\phi \nabla \phi \).

Step 2: Apply the divergence theorem

We then apply the divergence theorem to the given line integral \(\int_{C} \phi \nabla \phi \cdot n ds\). This transforms into a double integral over the region \(D\). That is, \(\int_{D} \nabla \cdot (\phi \nabla \phi ) dA\).

Step 3: Calculate the Divergence

Next, we will calculate the divergence of the functional vector \(\phi \nabla \phi\). After some calculation, we will find it to be \( \phi \nabla^{2} \phi + \nabla \phi \cdot \nabla \phi\) .

Step 4: Conclusion

We can see that \(\int_{D}\left(\phi \nabla^{2}\phi+\nabla \phi \cdot \nabla \phi\right) d A\) yelds the same result as the double integral obtained by applying divergence theorem. This proves the given identity.

Key Concepts

Divergence Theorem

The divergence theorem is a powerful tool in vector calculus which relates a surface integral over a closed surface to a volume integral over the region it encloses. It transforms complex surface integrals into much simpler volume integrals, making calculations more feasible. For a vector field \( \mathbf{F} \), the divergence theorem is expressed as:
\[ \int_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS = \int_V (abla \cdot \mathbf{F}) \, dV \]
where \( \partial V \) represents the boundary of volume \( V \) and \( \mathbf{n} \) is the outward facing unit normal vector on the surface. The term \( abla \cdot \mathbf{F} \) stands for the divergence of \( \mathbf{F} \), which is a measure of how much the vector field spreads out or converges at a certain point. In the context of our exercise, we applied the divergence theorem to the line integral given, which allowed the transformation from a line integral over curve \( C \) to a surface integral over the region \( D \) it encloses. Understanding this theorem is crucial for solving many problems in electromagnetism, fluid dynamics, and other fields of physics and engineering.

Line Integral

A line integral is a type of integral where a function is integrated along a curve. Specifically, in vector calculus, the line integral of a vector field \( \mathbf{F} \) along a curve \( C \) is defined as:
\[ \int_C \mathbf{F} \cdot d\mathbf{r} \]
It represents the work done by the vector field \( \mathbf{F} \) on moving a particle along the curve \( C \). In our exercise, the line integral is slightly different as it involves the vector field \( \mathbf{F} = \phi abla \phi \) and the dot product with the normal vector \( \mathbf{n} \) to the curve \( C \) with a scalar field \( \phi \). The question essentially asks to relate the line integral around a closed curve to quantities inside the domain it encompasses. In physical terms, if you imagine \( \phi \) as a scalar field like temperature or concentration, then the line integral is calculating the 'flux' of the gradient of \( \phi \) - how much of \( \phi \) is flowing out of the curve.

Vector Calculus

Vector calculus is an area of mathematics used to analyze mathematical fields that are multivariable and have direction and magnitude. It comprises various operations such as gradient, divergence, and curl, which are essential in the fields of physics and engineering. In the context of the exercise, the gradient \( abla \phi \) is a vector field representing the rate and direction of change of the scalar field \( \phi \) — for instance, the steepness and direction of a hill if \( \phi \) represented altitude. The divergence operation \( abla \cdot \mathbf{F} \) for a vector field \( \mathbf{F} \) measures the scalar quantity of a source or sink at a given point in a field. The more advanced operations like Laplacian \( abla^{2} \phi \) appear in our exercise as part of the proof. The Laplacian combines divergence and gradient operations and is used to describe the second-order behavior of \( \phi \) like the curvature or diffusion rate. Vector calculus is the language through which physical laws are described mathematically, seen in our context when converting line integrals to surface integrals using the divergence theorem.