# Determine the area, calculate the basis vectors and determine the inner product

• MHB
• Karl Karlsson1
In summary, the area that includes the point (x, y) = (0, 0) where the coordinate system is well defined is:\begin{align*}& A=\left( \frac {1}{y_0} \right)^2 + \left( \frac {1}{x_0} \right)^2\\&=\left( \frac {1}{y_0}+\frac {1}{x_0} \right)^2\\&=\left( \frac {1}{y_0}+\frac {x_0^2}{2} \right)^2\\&=\left(
Karl Karlsson1
A coordinate system with the coordinates s and t in $$\displaystyle R^2$$ is defined by the coordinate transformations: $$\displaystyle s = y/y_0$$ and $$\displaystyle t=y/y_0 - tan(x/x_0)$$ , where $$\displaystyle x_0$$ and $$\displaystyle y_0$$ are constants.

a) Determine the area that includes the point (x, y) = (0, 0) where the coordinate system
is well defined. Express the area both in the Cartesian coordinates (x, y) and in
the new coordinates (s, t).

b) Calculate the tangent basis vectors $$\displaystyle \vec E_s$$ and $$\displaystyle \vec E_t$$ and the dual basis vectors $$\displaystyle \vec E^s$$ and $$\displaystyle \vec E^t$$

c)Determine the inner products $$\displaystyle \vec E_s\cdot\vec E^s$$, $$\displaystyle \vec E_s\cdot\vec E^t$$, $$\displaystyle \vec E_t\cdot\vec E^s$$ and $$\displaystyle \vec E_t\cdot\vec E^t$$

My attempt:
a) Since $$\displaystyle tan(x/x_0)$$ is not defined for $$\displaystyle x=\pm\pi/2\cdot x_0$$ I assume x must be in between those values therefore $$\displaystyle -\pi/2\cdot x_0 < x < \pi/2\cdot x_0$$ and y can be any real number. Is this the correct answer on a)?

b) I can solve x and y for s and t which gives me $$\displaystyle y=y_0\cdot s$$ and $$\displaystyle x=x_0\cdot arctan(s-t)$$. $$\displaystyle \vec E_s = \frac {x_0} {1 + (s-t)^2}\cdot\vec e-x + y_0\cdot\vec e_y$$ and $$\displaystyle \vec E_t = - \frac { x_0} { 1 + (s-t)^2}\cdot\vec e_x$$. I get the dual basis vectors from $$\displaystyle \vec E^s = \frac {1} {y_0}\cdot\vec e_y$$ and $$\displaystyle \vec E^t = \frac {1} {y_0}\cdot\vec e_y - \frac {1} {x_0(1+(x/x_0)^2)}\cdot\vec e_x$$ , is this the correct approach?

c) It was here that I really started to question if i had done correct on a and b since I get $$\displaystyle \vec E_s\cdot \vec E^s = 1$$and$$\displaystyle \vec E_t\cdot \vec E^s = 0$$, this feels correct but then i get by just plugging in $$\displaystyle \vec E_t\cdot \vec E^t = \frac {x_0} {(1+(s-t)^2)(1+arctan(s-t)^2)}$$and $$\displaystyle \vec E_s\cdot \vec E^t = 1-\frac {1} {(1+(s-t)^2)(1+arctan(s-t)^2)}$$. Is this really correct? Because it feels like it is not correct.

a) Your answer for the area is correct. The point (0,0) is well-defined for any value of y_0, as long as x is within the given range.

b) Your approach for calculating the tangent basis vectors and dual basis vectors is correct. However, there is a slight mistake in your calculation for \vec E^t. It should be \vec E^t = \frac{1}{y_0}\cdot\vec e_y - \frac{1}{x_0(1+(s-t)^2)}\cdot\vec e_x. This will give you the correct answer for part c).

c) Your answers for \vec E_s\cdot\vec E^s and \vec E_t\cdot\vec E^s are correct. However, the calculation for \vec E_t\cdot\vec E^t is incorrect. The correct answer is \vec E_t\cdot\vec E^t = \frac{x_0}{(1+(s-t)^2)(1+(x/x_0)^2)}. And for \vec E_s\cdot\vec E^t, the correct answer is \vec E_s\cdot\vec E^t = -\frac{1}{(1+(s-t)^2)(1+(x/x_0)^2)}.

Overall, your approach and calculations are correct, but there were some small mistakes in your calculations for \vec E^t and \vec E_t\cdot\vec E^t. Keep up the good work!

## 1. What is the purpose of determining the area?

The purpose of determining the area is to quantify the amount of space occupied by a two-dimensional shape or surface. This information is useful in various fields such as mathematics, physics, and engineering.

## 2. How do you calculate the basis vectors?

To calculate the basis vectors, you first need to identify the two vectors that form the basis for the given shape or surface. Then, you can use the formula for finding the cross product of two vectors to calculate the basis vectors.

## 3. What is the inner product?

The inner product is a mathematical operation that takes two vectors and produces a scalar value. It is also known as the dot product and is calculated by multiplying the corresponding components of the two vectors and then adding the results.

## 4. Why is determining the area, calculating the basis vectors, and determining the inner product important?

These calculations are important because they provide valuable information about the geometric properties of a shape or surface. They can also be used to solve various mathematical problems and have applications in fields such as physics, engineering, and computer graphics.

## 5. Can these calculations be applied to three-dimensional shapes or surfaces?

Yes, these calculations can be extended to three-dimensional shapes and surfaces. In this case, you would need to determine the volume, calculate the basis vectors in three dimensions, and find the scalar triple product to determine the inner product.

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