25Please show the answer for each section clearly ### Vector Field Analysis Using Green's Theorem#### Problem StatementConsider the following region \( R \) and the vector field \( \mathbf{F} \):1. **Compute the two-dimensional divergence of the vector field.**2. **Evaluate both integrals in the flux form of Green's Theorem and check for consistency.**3. **Determine whether the vector field is source-free.**Given:\[ \mathbf{F} = (2xy, 9x^2 - y^2) \]\[ R \text{ is the region bounded by } y = x(6-x) \text{ and } y = 0. \]#### Steps and Calculations**a. The two-dimensional divergence is:**\[ \nabla \cdot \mathbf{F} = \_\_\_\_ \]**b. Set up the integral over the region:**\[\int \int_R \_\_\_\_ \, dy \, dx\]**Set up the line integral for the \( y = x(6 - x) \) boundary:**\[\int_0^6 \_\_\_\_ \, dt\]**Set up the line integral for the \( y = 0 \) boundary:**\[\int_0^6 \_\_\_\_ \, dt\]**Evaluate these integrals and check for consistency:**Select the correct choice below and fill in any answer boxes to complete your choice:**A.** The integrals are consistent because they all evaluate to \_\_\_\_.**B.** The integrals are not consistent because the integral over \( R \) evaluates to \_\_\_\_, while the line integrals evaluate to \_\_\_\_.**c. Is the vector field source-free?** ---### GuidanceTo evaluate the divergence:\[ \nabla \cdot \mathbf{F} = \frac{\partial (2xy)}{\partial x} + \frac{\partial (9x^2 - y^2)}{\partial y} \]Set up the double integral accordingly:\[ \int_{0}^{6} \int_{0}^{x(6-x)} (\nabla \cdot \mathbf{F}) \, dy \, dx \]For the line integrals, parameterize the boundaries and integrate \( \mathbf{F} \ **Integral Consistency Evaluation**Evaluate these integrals and check for consistency. Select the correct choice below and fill in any answer boxes to complete your choice.1. **Evaluate the Integrals** Choose one of the following: - \( \bigcirc \) **A.** The integrals are consistent because they all evaluate to \(\_\_\_\_\_\). - \( \bigcirc \) **B.** The integrals are not consistent because the integral over \( R \) evaluates to \(\_\_\_\_\_\) while the line integrals evaluate to \(\_\_\_\_\_\).2. **Is the Vector Field Source-Free?** Choose one of the following: - \( \bigcirc \) **A.** Yes, because the two-dimensional divergence is zero everywhere. - \( \bigcirc \) **B.** No, because the flux is zero only for the given region. - \( \bigcirc \) **C.** Yes, because the flux is zero for the given region. - \( \bigcirc \) **D.** No, because the two-dimensional divergence is not zero everywhere.This assessment requires the student to evaluate given integrals to determine their consistency and analyze the source-free nature of a vector field based on the divergence. Answer boxes are provided to input the evaluative results of integrals and line integrals following the theoretical analysis.

The given vector field is and the region R is given by and (a) To Find: The two-dimensional…

Consider the following region R and the vector field F. K a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green's Theorem and check for consistency. c. State whether the vector field is source free. F= (2xy,9x² - y2); R is the region bounded by y = x(6-x) and y = 0. a. The two-dimensional divergence is b. Set up the integral over the region. JSO dy dx Set up the line integral for the y=x(6-x) boundary. 6 jo 0 dt Set up the line integral for the y=0 boundary. 6 jo. dt 0 Evaluate these integrals and check for consistency. Select the correct choice below and fill in any answer boxes to complete your choice. OA. The integrals are consistent because they all evaluate to OB. The integrals are not consistent because the integral over R evaluates to c. Is the vector field source-free? while the line integrals evaluate to

Consider the following region R and the vector field F. K a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green's Theorem and check for consistency. c. State whether the vector field is source free. F= (2xy,9x² - y2); R is the region bounded by y = x(6-x) and y = 0. a. The two-dimensional divergence is b. Set up the integral over the region. JSO dy dx Set up the line integral for the y=x(6-x) boundary. 6 jo 0 dt Set up the line integral for the y=0 boundary. 6 jo. dt 0 Evaluate these integrals and check for consistency. Select the correct choice below and fill in any answer boxes to complete your choice. OA. The integrals are consistent because they all evaluate to OB. The integrals are not consistent because the integral over R evaluates to c. Is the vector field source-free? while the line integrals evaluate to

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Vectors In Two And Three Dimensions

Section9.FOM: Focus On Modeling: Vectors Fields

Problem 13P

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