91Ó°ÊÓ

A vector field assigns a vector to every point in space. These vectors can vary in magnitude and direction across different points. A common notation for vector fields is \( \mathbf{F}(x, y, z) \), where the components are usually functions of the spatial coordinates \( x, y, z \). In this exercise, the vector field is defined as \( \mathbf{F}(x, y, z) = (x+y) \mathbf{i} + z \mathbf{j} + xz \mathbf{k} \).The components of this vector field determine how vectors are distributed across a space. Here:

• The \( \mathbf{i} \) component \((x+y)\) suggests that the vector varies not only on the \( x \) component but also depends on the \( y \) position.
• The \( \mathbf{j} \) component is simply \( z \), meaning its influence linearly grows with the \( z \) coordinate.
• Lastly, the \( \mathbf{k} \) component, \( xz \), multiplies \( x \) with \( z \), indicating this component increases more rapidly in areas where both \( x \) or \( z \) are significant.

Understanding how vector fields behave helps in calculating properties like flux across a particular surface.

Flux calculation measures how much of the vector field passes through a surface. Imagine a surface in the vector field, like a mesh, and consider how much of the wind (as a vector field) flows through it. Flux can be simplified by using the Divergence Theorem.

The Divergence Theorem relates surface flux to a volume integral. It states that the total outflow of the vector field through a closed surface \(S\) can be determined by integrating the divergence over the volume \(V\) inside the surface:\[ \Phi = \int\int\int_V abla \cdot \mathbf{F} \, dV \]

In the current problem, after calculating the divergence \(abla \cdot \mathbf{F} = 1 + x\), it guides how flux is distributed within the enclosed volume of the surface.

The integral over a cubic volume involves evaluating an expression within the boundaries of a cube. Here, the cube extends from \(-1\) to \(1\) along each of the \( x, y, z \) axes. These boundaries make sure that each side of the cube is touching a face of the cube defined by these values.

To compute this, we perform successive integrations of the divergence \(1+x\):

• First, integrate with respect to \( z \), which captures all values from the bottom to the top face of the cube.
• Next, integrate the resulting expression in relation to \( y \), spanning from one vertical face to the opposite one.
• Finally, integrate with respect to \( x \), moving from the front to the back face.

By evaluating these integrals, we get the flux going through the entire cubic surface, resulting in a total flux of \(12\). This sequence efficiently accounts for every tiny part of the cube, supporting the divergence theorem's principle of equating surface and volume integrations.