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Suppose f⃗ (x,y)=(3x−4y)i⃗ +5xj⃗ and c is the counter-clockwise oriented sector of a circle centered at the origin with radius 3 and central angle π/3. use green's theorem to calculate the circulation of f⃗ around


c.

Respuesta :

Space

Answer:

[tex]\displaystyle \int_C {F \cdot} \, dr = \boxed{\bold{\frac{27 \pi}{2}}}[/tex]

General Formulas and Concepts:
Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:
[tex]\displaystyle \bold{(cu)' = cu'}[/tex]

Derivative Property [Addition/Subtraction]:
[tex]\displaystyle \bold{(u + v)' = u' + v'}[/tex]

Derivative Rule [Basic Power Rule]:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals

Integration Property [Multiplied Constant]:
[tex]\displaystyle \bold{\int {cf(x)} \, dx = c \int {f(x)} \, dx}[/tex]

Multivariable Calculus

Partial Derivatives

Vector Calculus (Line Integrals)

Circulation Density:
[tex]\displaystyle \bold{F = M \hat{\i} + N \hat{\j} \rightarrow \text{curl} \ \bold{F} \cdot \bold{k} = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}[/tex]

Green's Theorem [Circulation Curl/Tangential Form]:
[tex]\displaystyle \bold{\oint_C {F \cdot T} \, ds = \oint_C {M \, dx + N \, dy} = \iint_R {\bigg( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \bigg)} \, dx \, dy}[/tex]

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle F(x, y) = (3x - 4y) \hat{\i} + 5x \hat{\j}[/tex]

[tex]\displaystyle \text{Circle Region:} \ \left \{ {{r = 3} \atop {\theta = \frac{\pi}{3}}} \right.[/tex]

Step 2: Integrate Pt. 1

  1. Define vector functions M and N:
    [tex]\displaystyle M = 3x - 4y , \ N = 5x[/tex]
  2. [Circulation Density] Differentiate [Partial Derivatives and Derivative Rules + Properties]:
    [tex]\displaystyle \frac{\partial M}{\partial y} = -4 , \ \frac{\partial N}{\partial x} = 5[/tex]
  3. [Green's Theorem] Substitute in Circulation Density:
    [tex]\displaystyle \iint_R {\bigg( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \bigg)} \, dx \, dy = \iint_R {5 - (-4)} \, dA[/tex]
  4. [Integrals] Simplify [Integration Property - Multiplied Constant]:
    [tex]\displaystyle \int_C {F \cdot} \, dr = 9 \iint_R {} \, dA[/tex]

Step 3: Integrate Pt. 2

We can evaluate the Green's Theorem double integral we found using simple geometry techniques:

[tex]\displaystyle \begin{aligned}\int_C {F \cdot} \, dr & = 9 \iint_R {} \, dA \\& = 9 \bigg( \frac{\theta r^2}{2} \bigg) \\& = 9 \bigg( \frac{\frac{\pi}{3} (3)^2}{2} \bigg) \\& = \boxed{\bold{\frac{27 \pi}{2}}} \\\end{aligned}[/tex]

∴ we have calculated the circulation of F around C using Green's Theorem.

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Learn more about Green's Theorem: https://brainly.com/question/14237772

Learn more about multivariable calculus: https://brainly.com/question/17166987

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Topic: Multivariable Calculus

Unit: Green's Theorem and Surfaces

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