Question

Rotated Green's Theorem Use Stokes' Theorem to write the circulation form of Green's Theorem in the \(y z\) -plane.

Short answer

Based on the given step-by-step solution, the circulation form of Green's Theorem in the \(yz\)-plane using Stoke's Theorem can be written as: $$\oint_C (Qdy + Rdz) = \iint_S (\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})dA$$

Step-by-step solution

Review Green's Theorem and Stoke's Theorem

Green's Theorem is a fundamental result in vector calculus that relates a line integral around a closed curve C to the double integral of the curl of a vector field over the region D enclosed by the curve C. Let F be a C^1 vector field defined on an open set that contains D and C, then Green's Theorem can be written as: $$\oint_C (Pdx + Qdy) = \iint_D (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$$ Where F = (P, Q). The left side of the equation represents the circulation of the vector field around curve C, also known as circulation form. Stoke's Theorem, on the other hand, is a generalization of Green's Theorem to higher dimensions. It states that the line integral of a vector field around the boundary of a surface (closed curve) is equal to the surface integral of the curl of the vector field on the surface enclosed by the curve. Mathematically, Stoke's Theorem can be written as: $$\oint_C F \cdot d\mathbf{r}=\iint_S (\nabla \times F)\cdot d\mathbf{S}$$

Relating Green's Theorem to the curl of a vector field

Recall that for a three-dimensional vector field \(F = (P,Q,R)\) in \(xyz\)-plane, the curl of the vector field is given by: $$\nabla \times F=\langle \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \rangle$$ Considering only the \(yz\)-plane, we can assume that \(x = 0\), so any component involving \(x\) will be removed. Thus, our curl of the vector field F reduces to: $$\nabla \times F=\langle \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}, 0, 0 \rangle$$

Apply Stoke's Theorem in the \(yz\)-plane to find the circulation form of Green's Theorem

Now, we can write the circulation form of Green's Theorem in the \(yz\)-plane using Stoke's Theorem. For this, let our vector field F be a 3D vector field defined in the \(yz\)-plane, i.e., \(F = (0,Q(y,z),R(y,z))\). We can rewrite the Stoke's Theorem for the given vector field as: $$\oint_C F \cdot d\mathbf{r}=\iint_S (\nabla \times F)\cdot d\mathbf{S}$$ Substituting the curl of the vector field F in the \(yz\)-plane from Step 2: $$\oint_C F \cdot d\mathbf{r}=\iint_S \langle \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}, 0, 0 \rangle \cdot d\mathbf{S}$$ Let dC be the infinitesimal path tangent to curve C in the \(yz\)-plane, then we can write F · dC = Qdy + Rdz. Hence, the circulation form of Green's theorem in the \(yz\)-plane can be written as: $$\oint_C (Qdy + Rdz) = \iint_S (\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})dA$$ This is the desired circulation form of Green's Theorem in the \(yz\)-plane using Stoke's Theorem.

Key concepts

Stokes' Theorem

Stokes' Theorem is a powerful and elegant result in vector calculus that generalizes several theorems, including Green's Theorem, for higher dimensions. It bridges the concepts of surface and line integrals, providing a deep connection between the behavior of a vector field across a surface and around its boundary.

To understand Stokes' Theorem, picture a smooth surface 'S' with a boundary curve 'C'. This theorem states that the integral of the curl of a vector field \( \mathbf{F} \) over surface 'S' is equal to the line integral of \( \mathbf{F} \) around its boundary curve 'C'. In essence, Stokes' theorem enables us to transform a surface integral into a line integral, which can often simplify complex calculations.

Mathematical Formulation of Stokes' Theorem

Mathematically, Stokes' Theorem is expressed as:\[\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (abla \times \mathbf{F})\cdot d\mathbf{S}\]Here, \(d\mathbf{r} \) is a vector tangent to the boundary curve 'C', and \(d\mathbf{S} \) is a vector perpendicular to the surface 'S', representing an infinitesimal area on 'S'.The left-hand side of the equation is the circulation of the vector field \( \mathbf{F} \) around 'C', and the right-hand side is the curl flux of \( \mathbf{F} \) through 'S'. It's important to note that the theorem assumes 'C' is positively oriented relative to 'S', meaning that if you were to walk along 'C' in the direction dictated by the orientation, the surface would always be on your left.

Stokes' theorem is particularly useful when the curl of \( \mathbf{F} \) over 'S' is easier to evaluate than the circulation around 'C' or when 'S' is chosen such that \( abla \times \mathbf{F} \) simplifies nicely.

Circulation and Curl of a Vector Field

Circulation in the context of a vector field refers to the total amount of the field that rotates or 'circulates' around a path or loop. It gives a sense of how much a vector field twists or curls around a given curve. The concept of circulation is tightly linked to the physical idea of fluid flowing along the curve.

The curl is a vector that describes the infinitesimal rotation of the field at a given point in a vector field. For a three-dimensional vector field \( \mathbf{F} = (P, Q, R) \) defined in space, the curl is computed as:\[abla \times \mathbf{F}=\left\langle \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right\rangle\]
To visualize this, imagine a tiny paddle wheel placed in a flow described by the vector field; the way the paddle wheel would turn captures the essence of the curl at that point.

The connection between circulation and curl becomes apparent when we apply Stokes' Theorem. Through this theorem, the circulation around a closed loop 'C' can be related to the curl of the vector field across a surface 'S' bounded by 'C'. In essence, the circulation around 'C' is equal to the flux of the curl through 'S'. The remarkable thing about Stokes' Theorem is how it provides a method to calculate one (often complicated) quantity by converting it into another (often simpler) one.

Surface and Line Integrals

Surface and line integrals are fundamental concepts in multivariable calculus used to generalize the idea of integration to two and three dimensions, respectively.

A surface integral extends the concept of an integral over an interval to an integral over a two-dimensional surface. It is used to calculate quantities that spread out over a surface, such as the total mass of a thin sheet with varying density, or the electric flux through a surface in electromagnetism. Mathematically, for a vector field \(\mathbf{F}\), the surface integral is given by:\[\iint_S \mathbf{F} \cdot d\mathbf{S}\]where \(d\mathbf{S}\) is the infinitesimal area vector perpendicular to the surface 'S' at each point.

Understanding Line Integrals

A line integral, on the other hand, is carried out over a curve or path in space, and is often used to determine the work done by a force field (such as gravity or electromagnetism) on a particle moving along that curve. For a function \(f(x, y, z)\) or a vector field \(\mathbf{F}\), we express the line integral along a path 'C' as:\[\int_C f ds\quad \text{or}\quad \int_C \mathbf{F} \cdot d\mathbf{r}\]where 'ds' represents an infinitesimal length element along 'C' and \(d\mathbf{r}\) is the differential position vector tangential to 'C'.

In the context of Green's Theorem in the \(yz\)-plane, we're concerned with a special case of a line integral where the curve 'C' lies in the plane, and our vector field is pertinent to that plane. The careful application of such integrals is crucial for understanding fluid flow, magnetic fields, and many other physical phenomena.