Vector Calculus

Step 1: Find the tangent vector.
.

Step 2: Evaluate at .
.

Step 3: Convert to unit vector.
Magnitude .
Unit tangent .

Step 4: Conclusion.
The required unit tangent vector is .

Step 1: Parametrize the line.
The line from to can be written as , where varies from 0 to 2. Then, and .

Step 2: Evaluate the line integral.
Substitute : Hence, the line integral = 12.

Step 3: Verify calculation.
All substitutions are correct, and path is linear; thus, final answer is 12.

Final Answer: 12

Step 1: Evaluate divergence.
where is the Dirac delta function in three dimensions.

Step 2: Substitute into integral.

Step 3: Conclusion.
Thus, the value of the volume integral is

Step 1: Understanding the Concept:
This problem asks for the evaluation of a line integral of a vector field over a closed loop (a circle). A convenient method for solving such problems is to use Stokes' Theorem, which relates the line integral over a closed loop to the surface integral of the curl of the vector field over the surface enclosed by the loop.
Step 2: Key Formula or Approach:
Stokes' Theorem states: where C is the closed curve (the circle ) and S is the surface enclosed by C (the disk ).
The vector field is with , , and . The curl is given by .
Step 3: Detailed Explanation:
First, calculate the components of the curl: Now, calculate the curl: Next, we set up the surface integral. The surface S is the unit disk in the x-y plane. The differential surface element is for an anti-clockwise path. We can take the constant out of the integral: The integral is simply the area of the surface S, which is the area of a circle with radius . Area of S = . Finally, substitute the area back into the expression: Step 4: Final Answer:
The value of the integral is 1.