Question

Es sei durch \(\underline{f}(u, v)=\left[u, v, \frac{1}{4} u v\right]^{\top},|u| \leq 1,|v| \leq 1\) ein Flächenstück \(F\) gegeben. \(\overline{\text { Ferner seien }}\) $$ G(\underline{x})=1+\frac{1}{2} x y z \quad, \quad \underline{V}(\underline{x})=\left[\begin{array}{c} 1+z^{4} \\ 1+z^{4} \\ 1+x^{2} y^{2} \end{array}\right] \quad, \quad \text { mit } \underline{x}=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] $$ ein Skalar- und ein Vektorfeld auf \(\mathbb{R}^{3}\). Berechne (a) \(\int_{\boldsymbol{F}} G(\underline{x}) d \sigma\), (b) \(\int_{F} \underline{V}(\underline{x}) \cdot d \underline{\sigma}\).

Short answer

To provide the short answers for (a) and (b), compute the double integrals from steps 2 and 4.

Step-by-step solution

Calculate \( G(\underline{f}(u, v)) \) and \( det(J_{\underline{f}}) \)

First of all you have to calculate \( G(\underline{f}(u, v)) \) and \( det(J_{\underline{f}}) \) which is the determinant of the Jacobian matrix. Since the scalar field function \( G(\underline{x}) = 1 + 0.5xyz \), evaluate it at \( \underline{f}(u, v) \). Similarly, find the Jacobian matrix \( J_{\underline{f}} \) of the surface function \( \underline{f}(u, v) \) and calculate its determinant.

Compute the Flux Integral of Scalar Field

With \( G(\underline{f}(u, v)) \) and \( det(J_{\underline{f}}) \), substitute them into the formula for the flux integral for a scalar field: \(\int_{F} G(\underline{x}) d \sigma = \int_{D} G(\underline{f}(u, v)) det(J_{\underline{f}}) du dv \). Evaluate this double integral over the domain D defined by \( |u| \leq 1, |v| \leq 1 \).

Calculate \( \underline{V}(\underline{f}(u, v)) \) and \( (J_{\underline{f}})^T \)

The flux integral of a vector field involves the transpose of the Jacobian matrix as well as the vector field evaluated at the surface function. So, for the vector field \( \underline{V}(\underline{x}) \), evaluate it at \( \underline{f}(u, v) \). Then, find the Jacobian matrix \( J_{\underline{f}} \) of the surface function again and calculate its transpose \( (J_{\underline{f}})^T \).

Compute the Flux Integral of Vector Field

Now you have \( \underline{V}(\underline{f}(u, v)) \) and \( (J_{\underline{f}})^T \). Substitute them into the formula for the flux integral for a vector field: \( \int_{F} \underline{V}(\underline{x}) \cdot d \underline{\sigma} = \int_{D} \underline{V}(\underline{f}(u, v)) \cdot (J_{\underline{f}})^T du dv \). Evaluate this double integral over the domain D defined before.

Key concepts

Flux Integral

The concept of a Flux Integral may evoke images of particles flowing through a surface, and it’s not far off. In vector analysis, a flux integral quantifies the amount of a field passing through a given surface.

For scalar fields, the flux integral through a surface, F, is expressed as \[\int_{F} G(\underline{x}) d \sigma\], where \(G(\underline{x})\) is the scalar field's value at each point on F, and \(d \sigma\) represents the infinitesimal area element on the surface. In our exercise, we first calculate the value of \(G\) at the parametrized surface using \(\underline{f}(u, v)\) before integrating.

For vector fields, the flux integral looks slightly different. It's given by \[\int_{F} \underline{V}(\underline{x}) \cdot d \underline{\sigma}\], where \(\cdot\) denotes the dot product, and \(\underline{V}(\underline{x})\) is the vector field’s value at each point. This reflects how much of the field is flowing in the direction normal to the surface. Calculating this requires evaluating the vector field at the surface using \(\underline{f}(u, v)\) and then transforming the area element by using the transpose of the Jacobian matrix for correct orientation and magnitude.

Jacobian Matrix

The Jacobian Matrix is a bridge connecting different spaces, in this case, the parameter space and the surface in question. For a function that maps from one coordinate system to another, the Jacobian encapsulates how the mapping stretches or shrinks volumes.

In detail, the Jacobian for function \(\underline{f}(u, v)\) is a matrix containing all first-order partial derivatives. When we talk about the determinant of the Jacobian matrix, often abbreviated as \(det(J_{\underline{f}})\), it indicates the factor by which the function \(\underline{f}\) is scaling volumes in a small region around a point.

This scaling factor is essential in our exercise as it adjusts the infinitesimal areas in the parametric (u, v) domain to reflect the actual areas on the surface, F, making it possible to compute integrals over the surface in the context of the original coordinates. When dealing with the flux integral of a vector field, we concern ourselves with the transpose of the Jacobian matrix, \((J_{\underline{f}})^T\), to ensure the orientation of the transformed area element is consistent with the normal to the surface.

Scalar and Vector Fields

Navigating through fields is not just a farmer's concern—in mathematics, fields refer to Scalar and Vector Fields, each describing different types of phenomena. Scalar fields associate a single scalar value with every point in space, embodying concepts like temperature or pressure that have magnitude but no direction at each point.

In the exercise, \(G(\underline{x}) = 1 + \frac{1}{2} xyz\) is a scalar field. Any function providing a number for each point without indicating a specific direction fits the bill for a scalar field.

Conversely, vector fields assign a vector to each point, indicating both magnitude and direction—like wind velocities or magnetic field lines. The vector field in our exercise is \(\underline{V}(\underline{x})\), and its value at each point tells us how much and in which direction something (like a fluid or force) is moving.

Understanding these concepts is vital when solving integrals over surfaces, as they tell us what is flowing through the surface (in the case of scalar fields) or the flow’s direction and strength at each point of the surface (in the case of vector fields).