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Chapter 16: Problem 29

Green's first formula Suppose that \(f\) and \(g\) are scalar functions with continuous first- and second-order partial derivatives throughout a region \(D\) that is bounded by a closed piecewise-smooth surface \(S .\) Show that $$\iint_{S} f \nabla g \cdot \mathbf{n} d \sigma=\iiint_{D}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V$$ Equation (9) is Green's first formula. (Hint: Apply the Divergence Theorem to the field \(\mathbf{F}=f \nabla g . )\)

Short Answer

Expert verified
Apply the Divergence Theorem to show the equivalence of surface and volume integrals for \( f \nabla g \).

Step by step solution

01

State the Divergence Theorem

The Divergence Theorem states that for any vector field \( \mathbf{F} \) with continuous partial derivatives throughout a region \( D \) bounded by a closed surface \( S \), \( \iint_{S} \mathbf{F} \cdot \mathbf{n} \, d\sigma = \iiint_{D} abla \cdot \mathbf{F} \, dV \). This theorem relates surface integrals to volume integrals.
02

Define the Vector Field

Let \( \mathbf{F} = f abla g \) where \( f \) and \( g \) are scalar functions with continuous first- and second-order partial derivatives over the region \( D \). We will apply the Divergence Theorem to this vector field \( \mathbf{F} \).
03

Compute the Divergence of \( \mathbf{F} \)

Calculate the divergence of \( \mathbf{F} = f abla g \). By the product rule, \( abla \cdot \mathbf{F} = abla \cdot (f abla g) = (abla f) \cdot (abla g) + f abla^2 g \). Here, \( abla^2 g \) is the Laplacian of \( g \).
04

Apply Divergence Theorem to \( \mathbf{F} \)

Substitute the computed divergence into the Divergence Theorem equation: \( \iint_{S} f abla g \cdot \mathbf{n} \, d\sigma = \iiint_{D} (abla f \cdot abla g + f abla^2 g) \, dV \). Thus, we express the surface integral in terms of a volume integral.
05

Conclusion

We have shown that by applying the Divergence Theorem, \( \iint_{S} f abla g \cdot \mathbf{n} \, d\sigma = \iiint_{D} (f abla^2 g + abla f \cdot abla g) \, dV \), verifying Green's first formula.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divergence Theorem
The Divergence Theorem is a powerful tool that connects two types of integrals: surface integrals, which are calculated over a surface, and volume integrals, which are calculated over a volume. This theorem is essential for converting complex surface calculations into potentially simpler volume calculations. Let's break down its components:

  • **Vector Field**: A quantity that has both a magnitude and a direction at every point within a region. For the Divergence Theorem, we are interested in vector fields with continuous partial derivatives.
  • **Surface Integral**: This calculates the total "flux" through a surface. It requires a closed surface, meaning it must completely enclose a volume.
  • **Volume Integral**: This sums up a field throughout a three-dimensional space, helping us find total accumulation or loss inside a volume.
By applying the Divergence Theorem, we can seamlessly shift from studying what flows through a surface to what accumulates inside a volume. This translation is especially useful in physics and engineering, where understanding flow and accumulation is key.
Scalar Functions
A scalar function is a mathematical function that assigns a single value, or scalar, to each point in space, as opposed to a vector function which assigns vectors. Scalar functions are foundational in fields like vector calculus because they help express quantities such as temperature or pressure, which vary over a region:

  • **Continuous Derivatives**: In the context of calculus, we often require that these functions have continuous first- and second-order derivatives. This ensures that changes in the function are smooth.
  • **Gradient**: The gradient of a scalar function is a vector that points in the direction of the greatest rate of increase of the function. It's denoted by \( abla f \) and is pivotal in forming vector fields from scalar functions.
Scalar functions are crucial in the application of Green's first formula. By understanding the gradient and other properties of these functions, we can transform them into vector fields, enabling the application of the Divergence Theorem and other powerful mathematical tools.
Surface Integrals
Surface integrals extend the concept of integration to two-dimensional surfaces. Instead of integrating a function along a line or throughout a volume, surface integrals add up values over a surface in space. Here's what you need to know about them:

  • **Surface**: Must be a closed and piecewise-smooth boundary, which simply means it fully encloses some volume and is smooth apart from possibly a few sharp changes.
  • **Directionality**: These integrals need to choose a direction, typically outward, which aligns with the idea of flux, measuring how much of a field exits a surface.
  • **Application**: Surface integrals are employed in physics for calculating things like electric flux, which describes the flow of electric field across a surface.
Understanding surface integrals is essential when using the Divergence Theorem. It allows for a seamless transition from calculating the flux across a surface to finding the accumulation within a volume.
Volume Integrals
Volume integrals are an extension of single-variable integrals into three-dimensional space. They allow for the accumulation of quantities throughout a volume. Here are some key points:

  • **Three-Dimensional Region**: Volume integrals are calculated over a defined three-dimensional region, typically denoted by \( D \).
  • **Applications**: These integrals are frequently used in physics and engineering to compute things like mass, charge, or energy within a certain volume.
  • **Relating to Divergence**: In the context of the Divergence Theorem, evaluating a volume integral allows for determination of net outflow or inflow within the region by considering its divergence.
For Green's first formula, volume integrals play a pivotal role by allowing the conversion of the surface boundary's influence into a perspective of the entire volume's influence. Thus, mastering volume integrals is critical for delving into advanced vector calculus applications.

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Most popular questions from this chapter

In Exercises \(35-44,\) use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the given direction. Cylinder \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) outward through the portion of the cylinder \(x^{2}+y^{2}=1\) cut by the planes \(z=0\) and \(z=a\)

In Exercises \(27-34,\) integrate the given function over the given surface. Cone \(F(x, y, z)=z-x,\) over the cone \(z=\sqrt{x^{2}+y^{2}}\) \(0 \leq z \leq 1\)

In Exercises \(25-28\) , find the circulation and flux of the field \(F\) around and across the closed semicircular path that consists of the semicircular arch \(\mathbf{r}_{1}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq \pi,\) followed by the line segment \(\mathbf{r}_{2}(t)=t \mathbf{i},-a \leq t \leq a\) $$ \mathbf{F}=-y^{2} \mathbf{i}+x^{2} \mathbf{j} $$

In Exercises \(37-40, \mathbf{F}\) is the velocity field of a fluid flowing through a region in space. Find the flow along the given curve in the direction of increasing \(t .\) $$ \begin{array}{l}{\mathbf{F}=-4 x y \mathbf{i}+8 y \mathbf{j}+2 \mathbf{k}} \\\ {\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 2}\end{array} $$

The tangent plane at a point \(P_{0}\left(f\left(u_{0}, v_{0}\right), g\left(u_{0}, v_{0}\right), h\left(u_{0}, v_{0}\right)\right)\) on a parametrized surface \(\mathbf{r}(u, v)=f(u, v) \mathbf{i}+g(u, v) \mathbf{j}+h(u, v) \mathbf{k}\) is the plane through \(P_{0}\) normal to the vector \(\mathbf{r}_{u}\left(u_{0}, v_{0}\right) \times \mathbf{r}_{v}\left(u_{0}, v_{0}\right),\) the cross product of the tangent vectors \(\mathbf{r}_{u}\left(u_{0}, v_{0}\right)\) and \(\mathbf{r}_{v}\left(u_{0}, v_{0}\right)\) at \(P_{0} .\) In Exercises \(49-52,\) find an equation for the plane tangent to the surface at \(P_{0} .\) Then find a Cartesian equation for the surface and sketch the surface and tangent plane together. Hemisphere The hemisphere surface \(\mathbf{r}(\phi, \theta)=(4 \sin \phi \cos \theta) \mathbf{i}\) \(+(4 \sin \phi \sin \theta) \mathbf{j}+(4 \cos \phi) \mathbf{k}, 0 \leq \phi \leq \pi / 2,0 \leq \theta \leq 2 \pi,\) at the point \(P_{0}(\sqrt{2}, \sqrt{2}, 2 \sqrt{3}) \quad\) corresponding to \((\phi, \theta)=\) \((\pi / 6, \pi / 4)\)
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