91Ó°ÊÓ

First, let's define some key terms and symbols that will be used in Stokes' theorem: 1. Surface: A surface is a two-dimensional object embedded in three-dimensional space. In this context, we can imagine it as a piece of paper that can be curved or bent. 2. Boundary: The boundary of a surface is a closed curve that defines the edge of the surface. 3. Vector field: A vector field assigns a vector to each point in the space. In physics, it is useful to describe quantities that have both magnitude and direction at every point, such as velocity or force fields. 4. Curl: The curl of a vector field is a measure of the rotation or circulation of the field around a given point. Mathematically, the curl is a vector and is denoted as \(\nabla \times \vec{F}\), where \(\nabla\) is the del operator and \(\vec{F}\) is the vector field. 5. Line integral: The line integral of a vector field along a curve is a measure of the "work" done by the field in moving an object along that curve. It is denoted as \(\oint_C \vec{F} \cdot d\vec{r}\), where \(C\) is the boundary curve, and \(d\vec{r}\) is a small displacement along the curve. 6. Surface integral: The surface integral of the curl of a vector field over a surface is a measure of the total circulation of the field through that surface. It is denoted as \(\iint_S (\nabla \times \vec{F})\cdot d\vec{S}\), where \(S\) is the surface, and \(d\vec{S}\) is a small area element on the surface.

Stokes' Theorem states that the line integral of a vector field around the boundary of a smooth, oriented surface is equal to the surface integral of the curl of the vector field over the surface: $$ \oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F})\cdot d\vec{S} $$ The theorem demonstrates the connection between line integrals and surface integrals, showing that the circulation of a vector field around the boundary of a surface is equal to the total circulation through the interior of that surface.

Now, let's use Stokes' theorem to solve a problem. The given problem is: Find the line integral of the vector field \(\vec{F}(x,y,z) = (y^2, x^2, z^2)\) around the boundary of the surface \(S\), defined by \(z = 1 - x^2 - y^2\) and \(z \geq 0\). 1. Calculate the curl of the vector field: $$ \nabla \times \vec{F} = \left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ y^2 & x^2 & z^2 \end{array}\right| = \left(2z, 2y, -2x\right) $$ 2. Calculate the unit normal vector \(\hat{n}\) of the surface: Take the gradient of the function \(g(x,y,z) = 1 - x^2 - y^2 - z\) which defines the surface, then normalize it: $$ \nabla g = \left(-2x, -2y, -1\right) $$ $$ \hat{n} = \frac{\nabla g}{||\nabla g||} = \frac{1}{\sqrt{(-2x)^2 + (-2y)^2 + (-1)^2}}\left(-2x, -2y, -1\right) $$ 3. Calculate the surface integral of \((\nabla \times \vec{F})\cdot \hat{n}\) over the surface \(S\): $$ \iint_S (\nabla \times \vec{F})\cdot d\vec{S} = \iint_S (\nabla \times \vec{F})\cdot \hat{n} \ dS $$ Using polar coordinates, we can write \(x = r\cos{\theta}\) and \(y = r\sin{\theta}\), with \(r\) ranging from \(0\) to \(1\), and \(\theta\) ranging from \(0\) to \(2\pi\). Now, we can compute the surface integral: $$ \iint_S (\nabla \times \vec{F})\cdot \hat{n} \ dS = \int_{0}^{2\pi}\int_{0}^{1} (2(1 - r^2), 2r\sin{\theta}, -2r\cos{\theta})\cdot (-2r^2\cos{\theta}, -2r^2\sin{\theta}, -1) \ r drd\theta $$ 4. Evaluate the integral to find the line integral around the boundary: Solving the integral, we obtain $$ \oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F})\cdot d\vec{S} = \frac{4\pi}{3} $$ So, the line integral of the vector field \(\vec{F}(x,y,z) = (y^2, x^2, z^2)\) around the boundary of the surface \(S\) is equal to \(\frac{4\pi}{3}\).