Math Problem Statement
Evaluate, ∫F.dr where F = x²i + xyj and C is the boundary of square in the plane z=0 and bounded by the lines x=0, y=0, x=a & y=a
Solution
To evaluate the line integral , where and is the boundary of a square in the plane bounded by , , , and , we proceed as follows:
Step 1: Parameterize the boundary curve
The boundary of the square consists of four straight line segments. We will break the boundary curve into four parts and parameterize each one.
-
Segment 1 (from to ): The vector .
-
Segment 2 (from to ): The vector .
-
Segment 3 (from to ): The vector .
-
Segment 4 (from to ): The vector .
Step 2: Compute the line integral for each segment
Segment 1: ,
The line integral over this segment is:
Segment 2: ,
The line integral over this segment is:
Segment 3: ,
The line integral over this segment is:
Segment 4: ,
The line integral over this segment is:
Step 3: Add the contributions from all segments
Now we add up the integrals from all four segments:
Final Answer:
Would you like any more details or clarifications on the steps?
Here are some related questions to explore further:
- What is the line integral when the vector field is conservative?
- How do you apply Green's Theorem to this problem?
- Can you find the work done by a force field along a different path?
- What happens if the boundary is a circle instead of a square?
- How does the integral change if the square is rotated?
Tip: When solving line integrals, always pay attention to the direction of traversal along the boundary—incorrect parametrization can lead to sign errors.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Line Integrals
Vector Fields
Parameterization of Curves
Boundary Conditions
Formulas
∫F.dr = ∫ (F·dr)
Line integral formula for vector fields
Parameterization of curve segments
Theorems
Green's Theorem (as a related concept for line integrals)
Fundamental Theorem of Line Integrals
Suitable Grade Level
Grades 11-12 (Advanced Calculus/Vector Calculus)
Related Recommendation
Evaluate Line Integral of Vector Field F = x^2i + yj along x = y^2 from (4, 2) to (1, -1)
Evaluate Line Integral F • dr for a Vector Field on a Parametrized Curve
Line Integral of f(x, y, z) = x + √y - z^4 from (0,0,0) to (2,4,2)
Evaluate Line Integral along Curve C | Vector Fields and Parametrization
Evaluating Line Integral of Vector Field F = (y^2, x^2) Over Curve y = x - 1