This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Parameterize the line segment C. The line segment goes from P_0 = (-1, 2, 0) to P_1 = (3, 0, 1). A parameterization for a line segment is given by r(t) = P_0 + t(P_1 - P_0) for 0 t 1. P_1 - P_0 = (3 - (-1), 0 - 2, 1 - 0) = (4, -2, 1) So, the position vector r(t) is: r(t) = (-1, 2, 0) + t(4, -2, 1) r(t) = (-1 + 4t) i + (2 - 2t) j + t k This gives us the parametric equations for x, y, z: x(t) = -1 + 4t y(t) = 2 - 2t z(t) = t Step 2: Find dr in terms of t. Differentiate each component of r(t) with respect to t: (dx)/(dt) = (d)/(dt)(-1 + 4t) = 4 (dy)/(dt) = (d)/(dt)(2 - 2t) = -2 (dz)/(dt) = (d)/(dt)(t) = 1 So, dr = ( (dx)/(dt) i + (dy)/(dt) j + (dz)/(dt) k ) dt = (4 i - 2 j + 1 k) dt. Step 3: Express the vector field F in terms of t. The given vector field is F(x,y,z) = xz i - yz j + x k. Substitute x(t), y(t), and z(t) into F: F_x = xz = (-1 + 4t)(t) = -t + 4t^2 F_y = -yz = -(2 - 2t)(t) = -(2t - 2t^2) = -2t + 2t^2 F_z = x = -1 + 4t So, F(t) = (4t^2 - t) i + (2t^2 - 2t) j + (4t - 1) k. Step 4: Calculate the dot product F · dr. F · dr = (F_x (dx)/(dt) + F_y (dy)/(dt) + F_z (dz)/(dt)) dt F · dr = ( (4t^2 - t)(4) + (2t^2 - 2t)(-2) + (4t - 1)(1) ) dt F · dr = ( (16t^2 - 4t) + (-4t^2 + 4t) + (4t - 1) ) dt Combine like terms: F · dr = (16t^2 - 4t^2 - 4t + 4t + 4t - 1) dt F · dr = (12t^2 + 4t - 1) dt Step 5: Evaluate the definite integral. The integral is from t=0 to t=1: _C F · dr = _0^1 (12t^2 + 4t - 1) dt Integrate term by term: = [ (12t^3)/(3) + (4t^2)/(2) - t ]_0^1 = [ 4t^3 + 2t^2 - t ]_0^1 Now, evaluate at the limits: = ( 4(1)^3 + 2(1)^2 - 1 ) - ( 4(0)^3 + 2(0)^2 - 0 ) = (4 + 2 - 1) - (0) = 5 The value of the line integral is 5.