(a) Evaluate the line integral . F - dr where F : R → R³ is defined as F(x,y, 2) = (VI, –2 cos(y), rz) and c is the path given by e(t) = (t°, t, 18t²), t e [0, 1] %3D (b) Let F be a Cl vector field defined on an open, connected set U CR?. Precisely state 3 equivalent definitions for F being a gradient vector field. (Remember, you can reference your lecture notes for exams!)

Along the curve C traced out by the end point of the position vector tk (t) = (- sin ti + cos tj+t) = [0,π/2] calculate the line integral √ F(r) - di dr of the vector field F(x, y, z) = −y+xj+yzk.

For the vector field G = (yey + 4 cos(4x + y))i + (xeªy + cos(4x + y)) ♬ find the line integral of G along the curve C from the origin along the x-axis to the point (4,0) and then counterclockwise around the circumference of the circle x² + y² = 16 to the point (4/√√2,4/√√2). ScĞ·dr = 0 }

3. Evaluate |F. dr where C is the curve F(t) = (t, t°, t“), 0