6. A rock moves along the parabola x = y² + 2 under the force field F = (x², ye"). Find the work moving the rock from (2,0) to (3, 1).

Find the work done by the force field F(x, y) = x 2 i yex  on a particle that moves along the parabola x y 2 + 1 from (1, 0 ) to (2, 1)

What is the work done by a force field F (x, y) := 2x (20) to move a particle in a straight line from (0, 1) to (2,0)?

Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 5x² - 3xy + xyz (a) Find the rate of change of the potential at P(5, 2, 5) in the direction of the vector v = i + j - k. (b) In which direction does V change most rapidly at P? 73°F Mostly cloudy (c) What is the maximum rate of change at P? Need Help? Read It Show My Work (Optional)? Watch It Q

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Brazilian soccer star Marta has a penalty kick in the quarter-final match. She kicks the soccer ball from ground level with the (x, y)-coordinates (76, 21) on the soccer field shown in the figure and with initial velocity vo = 8i - 4j+23k ft/s. Assume an acceleration of 32 ft/s² due to gravity and that the goal net has a height of 8 ft and a total width of 24 ft. 105 ft 1 105 ft r₁ = rj = rk = 165 ft Determine the position function that gives the position of the ball t seconds after it is hit. (Use symbolic notation and fractions where needed.) r(t) = ri(t)i + rj(t)j + rk(t)k Incorrect 81+76 -41+21 ↓ 23r+1672² 165 ft 12 12

Please solve and show all steps / work clearly. Do not use a calculator. I will rate and like. Thank you.

Brazilian soccer star Marta has a penalty kick in the quarter-final match. She kicks the soccer ball from ground level with the (x, y)-coordinates (76, 21) on the soccer field shown in the figure and with initial velocity vo = 8i - 4j+23k ft/s. Assume an acceleration of 32 ft/s² due to gravity and that the goal net has a height of 8 ft and a total width of 24 ft. 105 ft 105 ft ri = rj = 165 ft rk = e 165 ft Determine the position function that gives the position of the ball t seconds after it is hit. (Use symbolic notation and fractions where needed.) r(t) = r(t)i + r(t)j +rk(t)k 12 12 X

6) Let f(xxx)=3xy²-x² be a potential function of a fluid velocity field. a) Find the magnitude of the greatest rate of change for f(x,y) at (1,3). Please provide a physical interpretation. b) Find the direction angles for this greatest rate of change. Express angles in degrees. Please provide a physical interpretation.

Find the work done by the force field F(x, y) = (x² In(y + 5), x3 -) on a particle 3(y + 5) that moves once around the circle x? + y? = 4 oriented in the counterclockwise direction.