91Ó°ÊÓ

A vector field is a function that assigns a vector to every point in space. Imagine it like a map of arrows, where each arrow shows both a direction and a magnitude at that point in space. In our exercise, we deal with the vector field \( \mathbf{F} = (5x^2, 5y^2, 5z^3) \). This means:

• The first component, \( 5x^2 \), indicates how the field behaves along the x-axis.
• The second component, \( 5y^2 \), shows the behavior along the y-axis.
• The third component, \( 5z^3 \), is about the z-axis.

Think of the vector field as a 3D force or flow across space, where each part \((x, y, z)\) gets assigned a unique vector \((5x^2, 5y^2, 5z^3)\). Understanding the behavior of vector fields is crucial when studying how they interact with surfaces and volumes in space.

A closed surface refers to a boundary that completely encloses a volume in three-dimensional space. Imagine a bubble; the surface of the bubble is closed because it wraps around the air inside it. In our exercise, the closed surface \( S \) is a sphere defined by the equation \( x^2 + y^2 + z^2 = 4 \), which is a perfect example:

• This equation depicts a sphere centered at the origin \((0, 0, 0)\).
• The radius of the sphere is 2 because \( x^2 + y^2 + z^2 = 4 \) suggests \( (x, y, z) \) is on a sphere with a radius of 2.
• Spheres are simple geometric shapes that encapsulate the volume within them.

Closed surfaces are essential in the divergence theorem because they enclose the volume for which we calculate both the surface integral and the volume integral.

A volume integral extends the concept of integration to three dimensions, allowing us to compute quantities over a 3D region. When using the divergence theorem, you convert a surface integral into a volume integral, which is often easier to compute.
In the exercise, we compute the volume integral of the divergence of a vector field over the sphere's volume. The divergence \( abla \cdot \mathbf{F} = 10x + 10y + 15z^2 \) tells us about how much the vector field spreads out from each point. Then, the integral \( \int \int \int_V abla \cdot \mathbf{F} \ dV \) gives:

• The total flux through the entire volume inside the sphere.
• This transformation leverages the fact that it is sometimes simpler to calculate within a volume than across its boundary.

Thus, instead of computing the flow over a complicated surface, transformation into a volume integral simplifies the math.

Spherical coordinates are a system of three numbers \((\rho, \phi, \theta)\) used to define positions in three-dimensional space. They are particularly useful for problems with symmetry around a point, like spheres. In this coordinate system:

• \( \rho \) is the radial distance from the origin to the point.
• \( \phi \) is the inclination angle from the positive z-axis.
• \( \theta \) is the azimuth angle in the xy-plane from the positive x-axis.

To solve the present problem using spherical coordinates, we transform \((x, y, z)\) as follows:

• \( x = \rho \sin \phi \cos \theta \).
• \( y = \rho \sin \phi \sin \theta \).
• \( z = \rho \cos \phi \).

This change of variables also modifies the volume element to \( \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \), making it easier to evaluate the integral over our spherical volume. Using spherical coordinates simplifies calculations that involve radial symmetry, such as those in this exercise.