91Ó°ÊÓ

A vector field is a mathematical description where a vector is assigned to every point in space. Imagine a map of wind patterns, each point showing wind's speed and direction. In our problem, the vector field is represented by \( \vec{F} = -y \vec{j} + z \vec{k} \). Here, it describes how vectors behave in three-dimensional space.
- \( \vec{j} \) and \( \vec{k} \) are unit vectors along the y and z axes, respectively.
- The term \( -y \vec{j} \) indicates that the vector field has a component in the negative y-direction, depending on the value of \( y \).
- The term \( z \vec{k} \) means there's also a z-direction component.
So, each point in space has a vector coming out of it that depends on y and z values. Understanding these vector components is critical for determining how they interact with surfaces in space, like \( S \), the surface in the task.

Surface parameterization is a handy way to describe a surface using parameters. This allows for complex surfaces to be explored easier by mapping them into known spaces. In this exercise, the surface given is expressed as \( z = y^2 + 5 \). We can parameterize this by letting \( \vec{r}(x, y) = \langle x, y, y^2 + 5 \rangle \).
- This means any point on the surface can be captured using the parameters \( x \) and \( y \).
- With the defined rectangle \( -2 \leq x \leq 1 \) and \( 0 \leq y \leq 1 \), we know the area we're evaluating the flux integral over.
By changing the coordinate system to a simpler parameter space, various calculations become easier, like finding normals and integrating over the surface.

Calculating the normal vector is a key step in finding flux, as it helps determine the proper orientation of the surface. For a parameterized surface \( \vec{r}(x, y) \), the normal vector \( \vec{n} \) is found using the cross product of the partial derivatives of \( \vec{r} \) with respect to the parameters.
- First, compute \( \frac{\partial \vec{r}}{\partial x} = \langle 1, 0, 0 \rangle \).
- Then, compute \( \frac{\partial \vec{r}}{\partial y} = \langle 0, 1, 2y \rangle \).
- The normal vector is then \( \vec{n} = \vec{r}_x \times \vec{r}_y = \langle 0, -2y, 1 \rangle \).
The cross product helps in finding the perpendicular vector to the surface at every point, which is integral in calculating the flux through the surface.

The double integral is used to calculate the flux of a vector field across a surface. In this context, it represents summing up all tiny bits of flux over a surface. To compute the flux integral \( \Phi \), use the formula \( \iint_S \vec{F} \cdot \vec{n} \, dS \).
- For our system, \( \vec{F} \cdot \vec{n} = 2y^2 + y^2 + 5 = 3y^2 + 5 \).
- First, integrate with respect to \( y \) over \( 0 \leq y \leq 1 \): \( \int_{0}^{1} (3y^2 + 5) \, dy \).
- Evaluate: \( \left[ y^3 + 5y \right]_0^1 = 6 \).
- Follow by integrating this result over the \( x \) range of \( -2 \leq x \leq 1 \): \( \int_{-2}^{1} 6 \, dx = 18 \).
The result, 18, is the amount of vector field flow through the surface \( S \), giving you a deeper understanding of the integral's purpose in these calculations.