Question

Green's First Identity Prove Green's First Identity for twice differentiable scalar-valued functions \(u\) and \(v\) defined on a region \(D:\) $$\iint_{R} u\left(f_{x}+g_{y}\right) d A=\oint_{C} u(\mathbf{F} \cdot \mathbf{n}) d s-\iint_{R}\left(f u_{x}+g u_{y}\right) d A$$ Show that with \(u=1\), one form of Green's Theorem appears. Which form of Green's Theorem is it?

Short answer

Question: Prove Green's First Identity for twice differentiable scalar-valued functions u and v, defined on a region D, and show that Green's Theorem can be derived using the special case when u=1. Answer: Using Green's Theorem and defining appropriate functions P(x, y) and Q(x, y), we were able to prove Green's First Identity: $$\oint_{C} u(\mathbf{F} \cdot \mathbf{n}) d s= \iint_{R} u\left(f_{x}+g_{y}\right) d A+\iint_{R}\left(f u_{x}+g u_{y}\right) d A$$ When setting u=1, we derived the divergence form of Green's Theorem: $$\oint_{C} (\mathbf{F} \cdot \mathbf{n}) d s= \iint_{R} (\nabla \cdot \mathbf{F})dA$$

Step-by-step solution

Write down given information

We are given that u and v are twice differentiable scalar-valued functions defined on a region D, and we need to prove Green's First Identity: $$\iint_{R} u\left(f_{x}+g_{y}\right) d A=\oint_{C} u(\mathbf{F} \cdot \mathbf{n}) d s-\iint_{R}\left(f u_{x}+g u_{y}\right) d A$$

Recall Green's Theorem

Green's Theorem states that for a continuously differentiable scalar function P(x, y) and a continuously differentiable scalar function Q(x, y), we have: $$\oint_{C} P dx + Q dy = \iint_{R} \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\, dA$$

Define P and Q

Now we need to define P(x, y) and Q(x, y) such that they relate to the given problem statement: $$P(x, y) = u(x, y)f(x, y),\quad Q(x, y) = -u(x, y)g(x, y)$$ Taking the partial derivatives, we get: $$\frac{\partial P}{\partial y} = u(x, y)f_y(x, y) + u_y(x, y)f(x, y)$$ $$\frac{\partial Q}{\partial x} = -u(x, y)g_x(x, y) - u_x(x, y)g(x, y)$$

Apply Green's Theorem

Now, plug P and Q into Green's Theorem: $$\oint_{C} \left[ u(x, y)f(x, y)dx - u(x, y)g(x, y)dy\right] = \iint_{R} \left[-u(x, y)g_x(x, y) - u_x(x, y)g(x, y) - u(x, y)f_y(x, y) - u_y(x, y)f(x, y)\right]dA$$

Identify Green's First Identity

Now, we can see that the result we just got is the desired Green's First Identity: $$\oint_{C} u(\mathbf{F} \cdot \mathbf{n}) d s= \iint_{R} u\left(f_{x}+g_{y}\right) d A+\iint_{R}\left(f u_{x}+g u_{y}\right) d A$$

Special case when u = 1

Now, let u = 1. Green's First Identity becomes: $$\oint_{C} (\mathbf{F} \cdot \mathbf{n}) d s= \iint_{R} \left(f_{x}+g_{y}\right) d A$$ This is a form of Green's Theorem.

Identify which form of Green's Theorem it is

With u = 1, the equation we derived above is actually the divergence form of Green's Theorem. In general, the divergence form of Green's Theorem is expressed as: $$\oint_{C} (\mathbf{F} \cdot \mathbf{n}) d s= \iint_{R} (\nabla \cdot \mathbf{F})dA$$ Here, \(\mathbf{F} = (f, g)\) and \(\nabla \cdot \mathbf{F} = f_x + g_y\), which matches our derived equation when u = 1.

Key concepts

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions of several variables. Unlike single-variable calculus, which deals with functions that depend on one variable, multivariable calculus handles functions that have two or more independent variables.

For instance, the function in Green's First Identity, involving twice differentiable scalar-valued functions like u and v, shines a spotlight on how we can manipulate and integrate these functions over higher-dimensional spaces. In our case, the region D represents a domain in the xy-plane where we perform integration.

The double integral represents accumulation over a two-dimensional area, and the line integral illustrates accumulation along a curve. These concepts are fundamental in describing and solving problems in fields that require accounting for variations across a plane, like physics, engineering, and economics.

Vector Calculus

Vector calculus is a branch of mathematics that deals with differentiation and integration of vector fields, primarily in 2-dimensional and 3-dimensional Euclidean spaces. It encompasses various operations involving vectors like dot product and cross product.

In the context of Green's First Identity, we encounter the dot product \(\textbf{F} \text{cdot} \textbf{n}\) which evaluates the component of a vector field \(\textbf{F}\) in the direction of a unit outward normal vector \(\textbf{n}\) along the boundary curve C. This is a typical operation in vector calculus where physical quantities like flux through a surface are calculated.

Vector calculus also introduces the gradient, divergence, and curl, which are differential operators that provide powerful ways to analyze the behavior of scalar and vector fields.

Green's Theorem

Green's Theorem is a fundamental statement in vector calculus that provides a relationship between a line integral around a simple closed curve C and a double integral over the plane region R bounded by C.

The theorem states that, under certain conditions, the circulation of a vector field around C is equal to the flux of the curl of the vector field across R. In simpler terms, it creates a bridge between the micro-local behavior of a field within a region and its macro-global behavior along the boundary.

Green's Theorem is actually a special case of the more general Stokes' Theorem, which applies to higher dimensions. It's worth noting that when we apply Green's Theorem with a function u equal to 1, we arrive at a simplified result that highlights the theorem's ability to transform a line integral into a double integral, thereby showcasing its utility in solving complex integral problems with greater ease and in obtaining important results related to the conservation of physical quantities.

Divergence Theorem

The divergence theorem, also known as Gauss's theorem, is another significant result in vector calculus, particularly useful in physics and engineering. This theorem connects the flow of a vector field through a closed surface to the behavior of the vector field inside the surface's boundary.

In mathematical terms, it relates the triple integral of the divergence of a vector field over a volume V to the flux of that field through the closed surface S that bounds V.

The form of Green's Theorem that appears when setting u to 1 in Green's First Identity is akin to the divergence theorem for a two-dimensional case. It's this intrinsic resemblance that helps us recognize the patterns of flux and divergence across different dimensions, thereby underscoring the versatility of vector calculus in describing and predicting the behavior of physical systems.