91Ó°ÊÓ

A vector field is a function that assigns a vector to each point in space. Imagine a field of arrows, each arrow having both a direction and a magnitude, pointing at distinct coordinates in a 3D space. For the given problem, we have a vector field defined as \( \mathbf{F}(x, y, z)=y^{2} \mathbf{i}+x \mathbf{j}+z^{2} \mathbf{k} \). This means at any point \((x, y, z)\), the vector field assigns a vector pointing in the direction determined by the components \( y^2, x, \text{ and } z^2 \).

• The \( \mathbf{i} \)-component depends on \( y^2 \), meaning it increases with the square of the y-coordinate.
• The \( \mathbf{j} \)-component is simply \( x \), corresponding directly to the x-coordinate.
• The \( \mathbf{k} \)-component is \( z^2 \), increasing with the square of the z-coordinate.

In Stokes' Theorem applications, vector fields allow us to relate the field's circulation around a loop to the field's behavior across a surface.

Surface integrals help us measure the field's effect over a surface, similar to how line integrals measure the effect along a curve. In Stokes' Theorem, the surface integral specifically involves the curl of a vector field over the surface.
For our problem, the surface \( S \) is part of a paraboloid, \( z = x^2 + y^2 \), truncated at \( z = 1 \).

• Given surface \( S \) is oriented upward, the surface normal points outward.
• We calculate \( abla \times \mathbf{F} \) to determine how the vector field swirls or rotates over this surface.
• The challenge lies in computing the integral of this curl over \( S \), achieving a relationship with the boundary integral.

By evaluating this integral, we understand the net rotational effect across the surface. This helps verify Stokes' Theorem by comparing it to the line integral around the boundary.

A line integral allows us to measure the work done by a vector field along a curve. In the context of Stokes’ Theorem, it calculates how the field circulates around the boundary of a given surface.
For this problem, the boundary curve \( \partial S \) is where the paraboloid cap meets the plane \( z = 1 \).

• We consider this circular boundary located at \( x^2 + y^2 = 1 \).
• Converted into polar coordinates: \( x = \cos(\theta) \), \( y = \sin(\theta) \), maintaining \( z = 1 \).
• The line integral of the vector field \( \mathbf{F} \) explores how \( \mathbf{F} \) aligns with this circular boundary.

By evaluating this expression, the integral quantifies the field’s circulation along the boundary, offering a means to verify Stokes' Theorem. If the line integral equals the surface integral, the field’s curl over the surface equals its circulation on the boundary.

The curl of a vector field is a vector operation that describes a field's tendency to rotate about a point. This concept is central to Stokes' Theorem as it links surface and line integrals.
In the given exercise, the vector field \( \mathbf{F}(x, y, z) = y^{2}\mathbf{i} + x\mathbf{j} + z^{2}\mathbf{k} \) has a curl, \( abla \times \mathbf{F} \), computed as follows:

• The \( \mathbf{i} \)-component: \(0\), as no \( \mathbf{k} \) or \( \mathbf{j} \) terms affect it.
• The \( \mathbf{j} \)-component: \(-1\), representing the field's rotational factor due to \( x \).
• The \( \mathbf{k} \)-component: \(-2y\), representing twisting from the \( y^2 \) term.

The curl provides a vector describing the rotational intensity and direction of the field locally. Integrating this over the surface captures the net twisting effect across \( S \). Understanding the curl helps determine the total circulation around \( \partial S \), essential for verifying Stokes’ relationship between integrals.