Friday, March 15, 2024

Functions of two variables

 Goal: recognize a function of two variables and identify its domain and range

Function of two variables

The definition of a function with two variables is similar to that of the function with one variable. The only difference is that we map a pair of two variables to another variable instead of mapping one variable to another variable.

Definition

A function of two variables z = f(x,y) maps each ordered pair (x,y) in a subset D of the real plane R² to a unique real number z. The set D is called the domain of the function. The range of f is the set of all real numbers z that has at least one ordered pair (x,y) 𝜺 D such that f(x,y) = z as shown in the following figure:




Examples                                                                                                                                                      


                                                                                                                                                  

Solution

a. This is an example of a linear function with two variables. There is no pair of variables (x, y) for which the function is not defined. Therefore the domain of the function is R².

To determine the range of the function, let's determine the set of reals z for which f(x,y) = z. We have 3x +5y + 2 = z. Let's solve this equation by choosing x = 0. We have 5y + 2 = z. y = z-2/5. The pair (0, z-2/5) is a solution of the equation 3x + 5y + 2 = z for any value of z. The range of the function is R.

b. For the function g to have a real value we need 9 - x² - y²≥ 0 or  - x² - y²≥ -9  x² + y² ≤ 9  

The domain D is defined as follow: D = {(x,y)ε R²/ x² + y² ≤ 9}.The graph of this set of points is described as a disk of radius 3. The graph includes the boundary as shown below.



    To determine the range of the function, we have to find out the set of reals for which g(x,y) = z.  The domain is made of circles starting from (0, 0) and ending at the boundary circle defined by x² + y² = 9. Let's find z for (0,0) i.e a point of the domain starting at the origin. We have g(0,0) = z. 

g(0,0) = ⎷9-(0)²-(0)² = ⎷9 = 3. Let's take a point of the boundary circle i.e (0,3). We have g(0,3) = ⎷9-(0)² - (3)⁰ = 0. The range is [0,3].

Practice




























 


Saturday, March 2, 2024

Arc length in polar curves (continued)

 In the last post, we set the formula for the arc length of a polar curve. Now let's do an example. 

Example


Solution

  When θ = 0, r = 2 + 2cos0 = 2 + 2 = 4.  As θ goes from 0 to 2ℼ, the cardioid is traced exactly once. Therefore 0 and  2ℼ represent the limits of integration. Using f(θ) = 2 + 2cos0 , ɑ = 0 and β = 2π , the formula for the arc length becomes:                                                                                                         
                                                                                                       

We have 1 + cosθ = 2cos²θ/2. Multiplying by 2: 2 + 2cosθ = 4cos²θ/2.                                         
                                                                                                                     
                                      

The absolute value is necessary because cosine is negative for some values of its domain. To resolve this issue, change the limit from 0 to ℼ and double the result. This strategy works because cosine is positive between 0 and ℼ/2 .                                                                                                               
 

Practice  

Find the arc length of r = 3sinθ                                                                                                                                           

                                                              :
 


                                                                                                                                                      
 

Saturday, February 24, 2024

Arc length in polar curves

Goal : find a formula for the arc length of a curve in polar coordinates

Arc length of a curve in polar coordinates

To find the formula for the arc length of a curve in polar coordinates, let's start from the formula of the arc length of a parametrized curve (x(t), y(t)) for a≤ t ≤b  in rectangular coordinates.




In polar coordinates the curve is defined by r = f(θ) and we also have:
x = rcosθ = f(θ)cosθ and y = rsinθ =f(θ)sinθ . Let's calculate dx/dθ and dy/dθ:



Let's replace dt by dθ and a and b by ɑ and β, which define the limits of integration of the curve in polar coordinates, in the formula for the arc length above:


This leads to the following theorem:

Theorem




 

Friday, February 23, 2024

Area between two polar curves

 Goal: find the area between 2 polar curves

Area between 2 polar curves

The procedure to find the area  between 2 polar curves is similar to that of the area of 2 curves in the system of coordinates in the cartesian plane, We find the points of intersection between the 2 curves and identify the functions that define the outer curve and the inner curve respectively.

Example

Find the area outside of the cardioid r = 2 + 2 sinθ and inside the circle r = 6 sinθ

Solution

 First, draw a graph containing both curves



To find the limits of integration, let's find the points of intersection by setting the 2 functions equal to each other and solving for θ



The solutions of this equation are θ = ℼ/6 and θ = 5ℼ/6, which are the limits of integration. The graph of the circle, in red.. is the outer curve. The graph of the cardioid, in blue, is the inner curve. To find the area between the 2 curves, let's subtract the area of the cardioid from that of the circle.



Practice

Find the area inside the circle r = 4 cosθ and outside the circle r = 2.

Tuesday, February 20, 2024

Area of a region bounded by a polar curve (continued)

 In the previous post, we set  the formula to find the area of a region bounded by a polar curve. Let's do an application.

Example. Find the area of one petal of the rose defined by the equation r = 3sin (2θ)

Solution

Here is the graph of the of the petal of the rose



The first petal of the rose is traced out from the polar coordinates (0, 0) and (0, 𝝅/2). To find the area inside the petal, let's use the formula of the area of the region bounded by a polar curve. In this formula we substitute 𝛼 by 0 and 𝛽 by 𝝅/2. 



To evaluate this integral, let's use the formula sin²𝛂 = 1 - cos(2𝛂) with 𝛂 = 2𝜃

Practice

Find the area inside the cardioid defined by the equation r = 1-cos𝜃

Saturday, February 17, 2024

Areas of regions bounded by polar curves

 To find the area of a region bounded by a curve in rectangular coordinates, we use the Riemann sum to approximate the area under the curve by using rectangles. In polar coordinates, we are going to use the Riemann sum also to find the area bounded by a curve but instead of using rectangles we will use sectors of a circle. Let's consider the curve defined by the function r = f(θ) where α ≤ θ ⩽ 𝛃. Our goal is to find the area bounded by the curve and the 2 radial lines θ = α and θ =𝛃. 




Let's start by dividing the area into sectors of equal width. We name the width Δθ and it's calculated by using this formula: Δθ = 𝛃 - α/n. Let's find the area of the sectors. They have equal area since their measurement is equal. The area of each sector is used to approximate the area between line segments. We sum the area of the sectors to approximate the total area. Let's find the formula for the area of a sector.

The area of a circle is given by A = 𝝅r². The length of a circle is 360 degrees or 2π. The surface for one radian is A = 𝝅r²/2π = r²/2. The area for a sector of Δθ radians is A =  Δθ r²/2. This represents the area of any sector. Let's call it Aᵢ and substitute r by f(θ). Aᵢ = 1/2 [f(θ)]².

Let's add the areas of all the sectors to approximate the area bounded by the polar curve and the radial lines :







Let's divide the sector in as many subintervals as possible. At some point we approach infinity. The area of the sector is then given by:

Theorem                                                                                                                                                   
                                                                                                                                                 
Suppose f is continuous and non negative on the interval  α ≤ θ ⩽ 𝛃 with 0 ≤ α - 𝛃 ≤ 2𝝅. The area of the region bounded by the graph r = f(θ) between the radial lines θ = α and θ = 𝛃   is:




                                                                                                                                                







Monday, February 12, 2024

Transforming polar equations into rectangular coordinates

 Transforming polar equations into rectangular coordinates leads to find a relation between x and y. To do so, we use the formulas that allow to convert points between coordinates.

Example

Rewrite each of the following equations into rectangular coordinates and identify the graph.

a. θ = π/3

b.  r = 3

c. r = 6cosθ-8sinθ

Solution

a,Let's take the tangent of of both sides:

tanӨ = tan𝝅/3 = ⎷3

Let's substitute tanӨ by y/x:

y/x = ⎷3   y = x⎷3. This is the equation of a straight line passing through the origin and of slope ⎷3. In general any polar equation of the form  θ = K represents a straight line passing through the pole and with slope tanK.

b. Let's use the equation x² + y² = r². Let's substitute r by 3:  x² + y² = 9. This is the equation of a circle centered at the origin. In general, any polar equation of the form r = k where k is a positive constant represents a circle centered at the origin and with radius k.

c.  Let's multiply both sides by r:

r² = 6rcosθ-8rsinθ.

Let's substitute rcosθ by x and rsinθ by y.

r² = 6x - 8y.

Let's use the equation x² + y² = r² and substitute r² :

x² + y² = 6x - 8y

x² -  6x + y² + 8y = 0

x² -  6x +9-9 + y² + 8y + 16-16 = 0

(x² -  6x +9) +(y² + 8y + 16) -25= 0

(x-3)² + (y + 4)² = 25

This the equation of a circle centered at the point (3, -4) with radius r = 5.

Practice

Rewrite the equation r = secθtanθ in rectangular coordinates and identify its graph.