This question is about gradients and directional derivatives. How can you tell the differences between these questions? Type 1: given that phi = x^2y^2z^2, find grad phi, grad phi at (-1,1,1) and a unit vector in this direction, the derivative of phi at (2,1,-1) in the direction of i and d = 3/5i + 4/5kThis question requires you to first find grad phi, plug in the coordinates and make a unit vector out of it, then multiply this unit vector by the two directions to get two solutions. Type 2: calculate the directional derivative of lamba(x) = x^2yz + 2xz^2 at the point (1,-2,1) in the direction of vector v=1/3(1,1,2) While, at first, it looks similar to the previous question, the working out is different. It requires you to find grad phi and sub in (1,-2,1). Then, you find the magnitude of the vector. You multiply grad phi at (1,-2,1) by the vector and divide all of that by the magnitude to give you the answer. I don't understand why it's different, Type 3: consider the scalar field phi = xz^2cos(x+y^2); find grad phi, a unit vector normal to the contour line of phi at (pi, 0, 1), the rate of change of phi at (pi, 0, 1) in the direction of j, the rate of change of phi at (pi, 0, 1) in the direction of the vector i + 2j - 2k. This one requires you to find grad phi, sub in the coordinates and find the unit vector of that to multiply j, similar to type 1, but for the last part, you add the values from the vector together to get 3 which turns into 1/3 that multiplies against the original directional vector of i + 2j - 2k and also the subbed in value from (pi, 0, 1). I need help understanding the difference between all these questions so I can recognise them when I see them.
This question is about gradients and directional derivatives. How can you tell the differences between these questions?
Type 1: given that phi = x^2y^2z^2, find grad phi, grad phi at (-1,1,1) and a unit
This question requires you to first find grad phi, plug in the coordinates and make a unit vector out of it, then multiply this unit vector by the two directions to get two solutions.
Type 2: calculate the directional derivative of lamba(x) = x^2yz + 2xz^2 at the point (1,-2,1) in the direction of vector v=1/3(1,1,2)
While, at first, it looks similar to the previous question, the working out is different. It requires you to find grad phi and sub in (1,-2,1). Then, you find the magnitude of the vector. You multiply grad phi at (1,-2,1) by the vector and divide all of that by the magnitude to give you the answer. I don't understand why it's different,
Type 3: consider the scalar field phi = xz^2cos(x+y^2); find grad phi, a unit vector normal to the contour line of phi at (pi, 0, 1), the rate of change of phi at (pi, 0, 1) in the direction of j, the rate of change of phi at (pi, 0, 1) in the direction of the vector i + 2j - 2k.
This one requires you to find grad phi, sub in the coordinates and find the unit vector of that to multiply j, similar to type 1, but for the last part, you add the values from the vector together to get 3 which turns into 1/3 that multiplies against the original directional vector of i + 2j - 2k and also the subbed in value from (pi, 0, 1).
I need help understanding the difference between all these questions so I can recognise them when I see them.
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