Math Problem Statement

1. What is the definition of a conservative vector field? 2. Let F(x, y) = P(x, y)i + Q(x, y)j be a vector field in the plane. What condition on the partial derivatives of P and Q guarantees that F is conservative? 3. State Green’s Theorem. Include a schematic diagram that explains how the boundary of the domain is oriented. 4. Define the curl of a vector field.

Solution

Here is a breakdown and explanation of each question:

1. Definition of a Conservative Vector Field

A vector field F\mathbf{F} is said to be conservative if it is the gradient of a scalar function, i.e., F=∇f\mathbf{F} = \nabla f, for some scalar function f(x,y)f(x, y). Equivalently:

  1. The line integral of F\mathbf{F} over any closed curve is zero: ∮CF⋅dr=0\oint_C \mathbf{F} \cdot d\mathbf{r} = 0.
  2. The line integral of F\mathbf{F} between two points is path-independent.

2. Condition on Partial Derivatives for Conservativeness

For a vector field F(x,y)=P(x,y)i+Q(x,y)j\mathbf{F}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j}, the condition for F\mathbf{F} to be conservative is: ∂P∂y=∂Q∂x.\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}. This is derived from the equality of mixed partial derivatives for the potential function f(x,y)f(x, y).

3. State Green's Theorem

Green's Theorem relates a line integral around a simple closed curve CC to a double integral over the plane region RR enclosed by CC. It states: ∮CF⋅dr=∬R(∂Q∂x−∂P∂y)dA,\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA, where F=P(x,y)i+Q(x,y)j\mathbf{F} = P(x, y)\mathbf{i} + Q(x, y)\mathbf{j}.

Orientation: The curve CC must be positively oriented (counterclockwise).

Diagram Suggestion: Include a simple closed curve with arrows indicating counterclockwise orientation, enclosing a region RR.

4. Definition of the Curl of a Vector Field

The curl of a vector field F(x,y,z)=P(x,y,z)i+Q(x,y,z)j+R(x,y,z)k\mathbf{F}(x, y, z) = P(x, y, z)\mathbf{i} + Q(x, y, z)\mathbf{j} + R(x, y, z)\mathbf{k} is a vector defined as:

\mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}.$$ In 2D ($$z = 0$$), the curl reduces to a scalar: $$\text{curl} \, \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}.$$ --- Do you want me to provide further explanations, examples, or diagrams for any of these? Let me know! ### Related Questions: 1. Can a non-conservative vector field become conservative under specific domain restrictions? 2. How does Green’s Theorem apply to computing areas of irregular regions? 3. Explain the physical meaning of the curl in 3D vector fields. 4. What conditions must be met for Green's Theorem to be applicable? 5. How do conservative fields relate to energy conservation in physics? **Tip:** To verify if a vector field is conservative, check for curl-free conditions and ensure the domain is simply connected!

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Math Problem Analysis

Mathematical Concepts

Conservative vector fields
Partial derivatives
Line integrals
Green's Theorem
Curl of a vector field

Formulas

Condition for conservativeness: ∂P/∂y = ∂Q/∂x
Green's Theorem: ∮C F · dr = ∬R (∂Q/∂x - ∂P/∂y) dA
Curl in 2D: curl F = ∂Q/∂x - ∂P/∂y

Theorems

Green's Theorem

Suitable Grade Level

Undergraduate (Calculus III)

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