Math Problem Statement
Consider the vector field
F
=⟨x
3
y
4
,x
4
y
3
⟩ Is this vector field Conservative? If so: Find a function f so that
F
=∇f f(x,y)=+k Use your answer to evaluate ∫
C
F ⋅d r along the curve C: r (t)=3cos(t) i +3sin(t) j ,0≤t≤ 4 π
Solution
To determine if the vector field is conservative and evaluate the given line integral, we follow these steps:
Step 1: Check if the field is conservative
A vector field is conservative if there exists a scalar potential function such that . This means:
The field is conservative if . For :
- , so ,
- , so .
Since , the field is conservative.
Step 2: Find the potential function
We find such that :
-
From , integrate with respect to : where is an arbitrary function of .
-
From , differentiate with respect to : Equating this to , we find , so is constant.
Thus, the potential function is: where is an arbitrary constant.
Step 3: Evaluate the line integral
Since is conservative, the line integral depends only on the potential difference between the endpoints of the curve :
The curve is , with . Substituting:
- Start point (): ,
- End point (): .
At both points, . Thus:
The line integral evaluates to:
Final Answer:
- The vector field is conservative.
- The potential function is .
- The line integral evaluates to 0.
Would you like further clarification or additional examples?
Related Questions:
- What is the general process to check if a vector field is conservative?
- How do you handle vector fields defined in three dimensions for conservativeness?
- Can conservative vector fields have closed-path integrals that are nonzero?
- How would the solution change if the curve was not closed?
- What is the physical significance of conservative vector fields?
Tip:
For closed curves in conservative vector fields, the line integral is always zero because the potential difference depends only on the endpoints!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Vector Fields
Conservative Fields
Line Integrals
Potential Functions
Formulas
Conservative field condition: ∂P/∂y = ∂Q/∂x
Gradient of a potential function: ∇f = F
Line integral in a conservative field: ∫C F ⋅ dr = f(x2, y2) - f(x1, y1)
Theorems
Fundamental Theorem of Line Integrals
Condition for Conservativeness of Vector Fields
Suitable Grade Level
Grades 11-12 and College
Related Recommendation
Evaluate Line Integral of Vector Field Along a Parametric Path
Determine if a Vector Field is Conservative and Find Its Potential Function
Calculate Line Integral of Vector Field F(x, y, z) along a Parameterized Curve
Determining if Vector Fields are Conservative or Not: Step-by-Step Solution
Determine Vector Field F⃗ from Line Integral ∫C F⃗ ⋅ dr⃗