Question

Under the (invertible) change of coordinates in \(E^{3}: x^{1}=\bar{x}^{1}, x^{2}=\bar{x}^{1}+\bar{x}^{2}, z^{3}=\bar{x}^{1}+\bar{x}^{3}\), a smooth path \(C: x^{i}=x^{i}(t), a \leq t \leq b\) and a smooth surface \(S: x^{i}=x^{i}(u, v), i=\) \(1,2,3,(u, v)\) in \(R_{u v}\), become \(\bar{C}\) and \(\bar{S}\) respectively. Let \(\bar{X}_{i}\) denote \(X_{i}\left(x^{1}, x^{2}, x^{3}\right)\) with \(x^{1}, x^{2}, x^{3}\) expressed in terms of \(\bar{x}^{1}, \bar{x}^{2}, \bar{x}^{3}\). a) Express the line integral \(\int_{C} X_{1} d x^{1}+X_{2} d x^{2}+X_{3} d x^{3}\) as an integral over \(\bar{C}\). b) Express the surface integral \(\iint_{S} X_{1} d x^{2} d x^{3}+X_{2} d x^{3} d x^{1}+X_{3} d x^{1} d x^{2}\) as a surface integral over \(\bar{S}\).

Short answer

Based on the above step-by-step solution, write a short answer summarizing the method and results. To find the line integral and surface integral in the new coordinate system, we performed the following steps: 1. We expressed the given coordinates (x, y, z) in the new coordinate system (∂x, ∂y, ∂z) using the given transformations. 2. We calculated the Jacobian matrix for the transformation, obtaining: $$J=\begin{pmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 0 & 1 \end{pmatrix}$$ 3. Using the transformed coordinates and Jacobian matrix, we re-expressed the line integral and surface integral in terms of the new coordinate system. By completing these steps, we successfully transformed the line integral and surface integral into the new coordinate system.

Step-by-step solution

Express the coordinates in the new coordinate system

First, we need to express the given coordinates \(x^{i}=x^{i}(t)\) and \(x^{i}=x^{i}(u, v)\) in terms of the new coordinate system \((\bar{x}^{1}, \bar{x}^{2},\bar{x}^{3})\). We are given the transformation: $$x^{1}=\bar{x}^{1}$$ $$x^{2}=\bar{x}^{1}+\bar{x}^{2}$$ $$x^{3}=\bar{x}^{1}+\bar{x}^{3}$$ Express the vector \(\mathbf{X}\) in terms of the new coordinates using these transformations: $$\bar{X}_{1}=X_{1}(x^{1}, x^{2}, x^{3})=X_{1}\left(\bar{x}^{1}, \bar{x}^{1}+\bar{x}^{2}, \bar{x}^{1}+\bar{x}^{3}\right)$$ $$\bar{X}_{2}=X_{2}(x^{1}, x^{2}, x^{3})=X_{2}\left(\bar{x}^{1}, \bar{x}^{1}+\bar{x}^{2}, \bar{x}^{1}+\bar{x}^{3}\right)$$ $$\bar{X}_{3}=X_{3}(x^{1}, x^{2}, x^{3})=X_{3}\left(\bar{x}^{1}, \bar{x}^{1}+\bar{x}^{2}, \bar{x}^{1}+\bar{x}^{3}\right)$$ With these expressions, we can now move on to find the Jacobian matrix.

Find the Jacobian matrix for the transformation

The Jacobian matrix \(J\) for the given transformation is defined as follows: $$J=\begin{pmatrix} \frac{\partial x^{1}}{\partial \bar{x}^{1}} & \frac{\partial x^{1}}{\partial \bar{x}^{2}} & \frac{\partial x^{1}}{\partial \bar{x}^{3}}\\ \frac{\partial x^{2}}{\partial \bar{x}^{1}} & \frac{\partial x^{2}}{\partial \bar{x}^{2}} & \frac{\partial x^{2}}{\partial \bar{x}^{3}}\\ \frac{\partial x^{3}}{\partial \bar{x}^{1}} & \frac{\partial x^{3}}{\partial \bar{x}^{2}} & \frac{\partial x^{3}}{\partial \bar{x}^{3}} \end{pmatrix}$$ By calculating the partial derivatives, we get: $$J=\begin{pmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 0 & 1 \end{pmatrix}$$ Now, we can use this Jacobian matrix to transform the line integral and surface integral.

Step 3a: Calculate the line integral in the new coordinate system

To compute the line integral, let's first differentiate the paths \(\bar{C}\) and \(C\) with respect to \(t\): $$\frac{dx^1}{dt} = \frac{d(\bar{x}^1)}{dt}$$ $$\frac{dx^2}{dt} = \frac{d(\bar{x}^1 + \bar{x}^2)}{dt}$$ $$\frac{dx^3}{dt} = \frac{d(\bar{x}^1 + \bar{x}^3)}{dt}$$ Now we can express the integral over the path \(\bar{C}\): $$\int_C \bar{X}_{1} dx^{1} + \bar{X}_{2} dx^{2} + \bar{X}_{3} dx^{3} = \int_{\bar{C}} \left(\bar{X}_{1} \frac{dx^1}{dt} + \bar{X}_{2} \frac{dx^2}{dt} + \bar{X}_{3} \frac{dx^3}{dt}\right) dt$$

Step 3b: Calculate the surface integral in the new coordinate system

To compute the surface integral, let's take a matrix determinant to express the differential area element \(dA\) using the Jacobian matrix: $$\begin{vmatrix} \frac{\partial x^{1}}{\partial u} & \frac{\partial x^{1}}{\partial v} & d{x}^{1}\\ \frac{\partial x^{2}}{\partial u} & \frac{\partial x^{2}}{\partial v} & d{x}^{2}\\ \frac{\partial x^{3}}{\partial u} & \frac{\partial x^{3}}{\partial v} & d{x}^{3} \end{vmatrix} = \begin{vmatrix} \frac{\partial \bar{x}^{1}}{\partial u} & \frac{\partial \bar{x}^{1}}{\partial v} & 1\\ \frac{\partial (\bar{x}^{1}+\bar{x}^{2})}{\partial u} & \frac{\partial (\bar{x}^{1}+\bar{x}^{2})}{\partial v} & 1\\ \frac{\partial (\bar{x}^{1}+\bar{x}^{3})}{\partial u} & \frac{\partial (\bar{x}^{1}+\bar{x}^{3})}{\partial v} & 1 \end{vmatrix}$$ Now, express the surface integral over the surface \(\bar{S}\): $$\iint_{S} X_{1} dx^{2} dx^{3}+X_{2} dx^{3} dx^{1}+X_{3} dx^{1} dx^{2} = \iint_{\bar{S}} \left(\bar{X}_{1} d\bar{x}^{2} d\bar{x}^{3}+\bar{X}_{2} d\bar{x}^{3} d\bar{x}^{1}+\bar{X}_{3} d\bar{x}^{1} d\bar{x}^{2}\right) dA$$ We have now transformed both the line integral and surface integral in the new coordinate system \((\bar{x}^{1}, \bar{x}^{2},\bar{x}^{3})\) as required by the exercise.

Key concepts

Coordinate Transformation

Coordinate transformation is a powerful technique in multivariable calculus that allows us to simplify complex problems by changing the coordinate system in which equations are expressed. Imagine having a set of equations that appear complicated due to their existing coordinate system. By transforming coordinates, we effectively change the perspective from which we view the problem, often simplifying the equations and making them easier to solve.

For example, if we have three coordinates \(x^1, x^2, x^3\) and transform them into \(\bar{x}^1, \bar{x}^2, \bar{x}^3\) using given transformation rules like \(x^1 = \bar{x}^1\), \(x^2 = \bar{x}^1 + \bar{x}^2\), and \(x^3 = \bar{x}^1 + \bar{x}^3\), we create a new, often more manageable way to look at the problem. This transformation can make the calculations easier and more intuitive.

The main goal of coordinate transformation is to find a representation that aligns more closely with the given problem's geometry or symmetry.

Line Integrals

Line integrals are a type of integral that allow us to integrate a function along a curve or path. Unlike regular integrals that operate over a straight line or axis, line integrals evaluate a function along a specific path in space, which can be curved or straight.

This is particularly useful in physics and engineering where we need to calculate quantities like work done by a force field. In our exercise, the line integral \( \int_{C} X_{1} dx^1 + X_{2} dx^2 + X_{3} dx^3 \) is transformed over the new path \(\bar{C}\), using the transformed coordinates.

To perform a line integral, we typically:

• Parametrize the path using a parameter, often \(t\).
• Express the function in terms of the parameter.
• Integrate over the parameter from the start to the end of the path.

By applying transformations, we can simplify the integral calculations by aligning them with the new coordinate system.

Surface Integrals

Surface integrals extend the concept of line integrals to functions over a surface. They allow us to compute the integral of a scalar field or vector field over a surface in three dimensions. Surface integrals are crucial in fields like fluid dynamics and electromagnetism, where we want to calculate flow across a surface or field strength on a membrane.

In the exercise, we transformed the surface integral \( \iint_{S} X_{1} dx^{2} dx^{3} + X_{2} dx^{3} dx^{1} + X_{3} dx^{1} dx^{2} \) to be evaluated over the new surface \(\bar{S}\).

Surface integrals involve:

• Parameterizing the surface using two parameters, often \(u\) and \(v\).
• Expressing all functions in terms of these parameters.
• Integrating over the parameter domain.

Transformations like the Jacobian matrix help us understand how area elements change, ensuring our integrals reflect the correct geometry.

Jacobian Matrix

The Jacobian matrix is a fundamental tool for relating different coordinate systems, especially when dealing with transformations. It is a square matrix of all first-order partial derivatives of a vector function. In transformations, the Jacobian gives us a way to relate volumes, areas, or lengths in different coordinate systems, making it essential for integrating after a transformation.

For instance, when transforming from \(x^1, x^2, x^3\) to \(\bar{x}^1, \bar{x}^2, \bar{x}^3\), the Jacobian matrix captures how small changes in one set of coordinates affect the other.

The Jacobian matrix is crucial for:

• Determining how differential elements (like area or volume) transform under a change of coordinates.
• Ensuring the transformed integrals accurately reflect the new geometry.
• Simplifying the computation of line and surface integrals after transformation.

In our exercise, the matrix was used to help transform both line and surface integrals into their new coordinate framework, ensuring we can integrate over the transformed path and surface with accuracy.