Worksheet: The Gradient Vector Field

In this worksheet, we will practice examining gradient vector fields produced from a multivariate function.

Q1:

Compute the gradient of the function 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 + 5 𝑦 .

  • A 1 2 , 1 5
  • B 5 , 2
  • C 1 5 , 1 2
  • D 2 , 5
  • E 2 𝑥 , 5 𝑦

Q2:

Find the gradient of 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦 1 .

  • A 2 𝑥 1 , 2 𝑦 1
  • B 2 𝑦 , 2 𝑥
  • C 2 𝑦 1 , 2 𝑥 1
  • D 2 𝑥 , 2 𝑦
  • E 𝑥 , 𝑦

Q3:

Find the gradient of 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 . l n

  • A 1 𝑥 𝑦 , 1 𝑥 𝑦
  • B 1 𝑦 , 1 𝑥
  • C ( 𝑥 , 𝑦 )
  • D 1 𝑥 , 1 𝑦
  • E ( 𝑥 , 𝑦 ) l n l n

Q4:

Compute the gradient of the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑒 2 𝑦 .

  • A 2 𝑥 𝑒 , 𝑦 𝑥 𝑒 𝑦 2 𝑦
  • B 𝑥 𝑒 , 2 𝑥 𝑒 2 𝑦 𝑦
  • C ( 2 𝑒 , 𝑥 𝑒 ) 𝑦 𝑦
  • D 2 𝑥 𝑒 , 𝑥 𝑒 𝑦 2 𝑦
  • E ( 𝑥 𝑒 , 2 𝑒 ) 𝑦 𝑦

Q5:

Find the gradient of 𝑓 ( 𝑥 , 𝑦 ) = 1 𝑥 + 𝑦 .

  • A 2 𝑥 ( 𝑥 + 𝑦 ) , 2 𝑦 ( 𝑥 + 𝑦 )
  • B 2 𝑥 ( 𝑥 + 𝑦 ) , 2 𝑦 ( 𝑥 + 𝑦 )
  • C 2 𝑥 ( 𝑥 + 𝑦 ) , 2 𝑦 ( 𝑥 + 𝑦 )
  • D 2 𝑥 ( 𝑥 + 𝑦 ) , 2 𝑦 ( 𝑥 + 𝑦 )
  • E 𝑥 ( 𝑥 + 𝑦 ) , 𝑦 ( 𝑥 + 𝑦 )

Q6:

Compute the gradient for 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 + 𝑧 .

  • A 2 𝑥 , 2 𝑧 , 2 𝑦
  • B 2 𝑦 , 2 𝑥 , 2 𝑧
  • C 2 , 2 , 2
  • D 2 𝑥 , 2 𝑦 , 2 𝑧
  • E 𝑥 , 𝑦 , 𝑧

Q7:

Compute the gradient for 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 𝑒 .

  • A 𝑥 𝑧 𝑒 , 𝑥 𝑦 𝑒 , 2 𝑥 𝑒
  • B 𝑥 𝑧 𝑒 , 2 𝑥 𝑒 , 𝑥 𝑦 𝑒
  • C 𝑥 𝑦 𝑒 , 𝑥 𝑧 𝑒 , 2 𝑥 𝑒
  • D 2 𝑥 𝑒 , 𝑥 𝑧 𝑒 , 𝑥 𝑦 𝑒
  • E 2 𝑦 𝑒 , 𝑦 𝑧 𝑒 , 𝑦 𝑒

Q8:

Compute the gradient for 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 𝑦 𝑧 . s i n

  • A ( 𝑦 𝑧 𝑥 𝑦 𝑧 , 𝑥 𝑦 𝑥 𝑦 𝑧 , 𝑥 𝑧 𝑥 𝑦 𝑧 ) c o s c o s c o s
  • B ( 𝑥 𝑧 𝑥 𝑦 𝑧 , 𝑦 𝑧 𝑥 𝑦 𝑧 , 𝑥 𝑦 𝑥 𝑦 𝑧 ) c o s c o s c o s
  • C ( 𝑥 𝑧 , 𝑦 𝑧 , 𝑥 𝑦 )
  • D ( 𝑦 𝑧 𝑥 𝑦 𝑧 , 𝑥 𝑧 𝑥 𝑦 𝑧 , 𝑥 𝑦 𝑥 𝑦 𝑧 ) c o s c o s c o s
  • E ( 𝑦 𝑧 , 𝑥 𝑧 , 𝑥 𝑦 )

Q9:

Compute the gradient for 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 + 𝑧 . 2 2 2

  • A 𝑥 𝑥 + 𝑦 + 𝑧 , 𝑧 𝑥 + 𝑦 + 𝑧 , 𝑦 𝑥 + 𝑦 + 𝑧 2 2 2 2 2 2 2 2 2
  • B 𝑦 𝑥 + 𝑦 + 𝑧 , 𝑥 𝑥 + 𝑦 + 𝑧 , 𝑧 𝑥 + 𝑦 + 𝑧 2 2 2 2 2 2 2 2 2
  • C 1 𝑥 + 𝑦 + 𝑧 , 1 𝑥 + 𝑦 + 𝑧 , 1 𝑥 + 𝑦 + 𝑧 2 2 2 2 2 2 2 2 2
  • D 𝑥 𝑥 + 𝑦 + 𝑧 , 𝑦 𝑥 + 𝑦 + 𝑧 , 𝑧 𝑥 + 𝑦 + 𝑧 2 2 2 2 2 2 2 2 2
  • E 𝑥 𝑥 + 𝑦 + 𝑧 , 𝑦 𝑥 + 𝑦 + 𝑧 , 𝑧 𝑥 + 𝑦 + 𝑧 2 2 2 2 2 2 2 2 2

Q10:

Find a function so that the vector field is a gradient field.

  • A
  • B
  • C
  • D
  • E

Q11:

We explore an example where a vector field F = 𝐹 , 𝐹 1 2 satisfies 𝜕 𝐹 𝜕 𝑦 𝜕 𝐹 𝜕 𝑥 = 0 1 2 but does not come from a potential. On the plane with the origin removed, consider the vector field F ( 𝑥 , 𝑦 ) = 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦 2 2 2 2 .

On the (open) half-plane 𝑥 > 0 , we can define the angle function 𝜃 ( 𝑥 , 𝑦 ) = 𝑦 𝑥 a r c t a n . This is well defined and gives a value between 𝜋 2 and 𝜋 2 . What is the gradient 𝜃 ?

  • A 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦 2 2 2 2
  • B 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦 2 2 2 2
  • C 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦 2 2 2 2
  • D 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦 2 2 2 2
  • E 𝑥 𝑥 + 𝑦 , 𝑦 𝑥 + 𝑦 2 2 2 2

Using the figure shown, use 𝜃 above to define the angle function 𝜃 ( 𝑥 , 𝑦 ) 1 on the region 𝑦 > 0 by a suitable composition with a 𝜋 2 rotation.

  • A 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( 𝑦 , 𝑥 ) + 𝜋 2 1
  • B 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( 𝑦 , 𝑥 ) 1
  • C 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( 𝑦 , 𝑥 ) + 𝜋 2 1
  • D 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( 𝑦 , 𝑥 ) 1
  • E 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( 𝑦 , 𝑥 ) + 𝜋 2 1

What is 𝜃 ( 𝑥 , 𝑦 ) 1 ?

  • A 𝑥 𝑥 + 𝑦 , 𝑦 𝑥 + 𝑦 2 2 2 2
  • B 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦 2 2 2 2
  • C 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦 2 2 2 2
  • D 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦 2 2 2 2
  • E 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦 2 2 2 2

Since 𝜃 and 𝜃 1 agree on the quadrant 𝑥 > 0 , 𝑦 > 0 , we can define the angle 𝑇 ( 𝑥 , 𝑦 ) at each point of the union with values between 𝜋 2 and 3 𝜋 2 . Using this, what is 𝐶 F r d , where 𝐶 is the arc of the unit circle from 1 2 , 3 2 to 1 2 , 1 2 ? Answer in terms of 𝜋 .

  • A 1 3 𝜋 1 2
  • B 1 2 𝜋 1 3
  • C 𝜋 1 2
  • D 3 𝜋 5
  • E 𝜋 5

In the same way, we can define 𝜃 2 on the half-plane 𝑥 < 0 and 𝜃 3 on 𝑦 < 0 . Hence, evaluate the line integral 𝐶 F r d around the circle of radius 2 , starting from 𝑃 ( 1 , 1 ) and going counterclockwise back to 𝑃 , stating your answer in terms of 𝜋 .

  • A 2 𝜋
  • B 𝜋 2
  • C 𝜋
  • D 2 𝜋
  • E 𝜋 2

Q12:

For 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑧 𝑥 + 𝑦 2 2 in Cartesian coordinates, find 𝑓 in cylindrical coordinates.

  • A 2 𝑧 𝑟 𝑒 + 𝑧 𝑟 𝑒 3 𝑟 2 𝑧
  • B 2 𝑧 𝑟 𝑒 + 1 𝑟 𝑒 3 𝑟 2 𝑧
  • C 2 𝑧 𝑟 𝑒 + 𝑧 𝑟 𝑒 3 𝑟 2 𝑧
  • D 2 𝑧 𝑟 𝑒 + 1 𝑟 𝑒 3 𝑟 2 𝑧
  • E 2 𝑧 𝑟 𝑒 𝑧 𝑟 𝑒 3 𝑟 2 𝑧

Q13:

Suppose 𝑤 = 𝐹 ( 𝜙 ( 𝑥 , 𝑦 ) ) with 𝜙 = ( 𝑥 + 𝑦 , 𝑥 𝑦 , 𝑥 𝑦 ) . Express the gradient 𝑤 𝜋 , 2 3 (viewed as a 1 × 2 matrix) in terms of the 1 × 3 matrix 𝐹 ( 𝑞 ) , where 𝑞 = 𝜙 𝜋 , 2 3 , and a matrix of partial derivatives of 𝜙 .

  • A 𝑤 𝜋 , 2 3 = 𝐹 ( 𝑞 ) 𝜋 2 3 𝜋 2 3 2 3 𝜋
  • B 𝑤 𝜋 , 2 3 = 𝐹 ( 𝑞 ) 2 𝜋 4 3 2 𝜋 4 3 2 3 𝜋
  • C 𝑤 𝜋 , 2 3 = 𝐹 ( 𝑞 ) 2 𝜋 4 3 2 𝜋 4 3 2 3 𝜋
  • D 𝑤 𝜋 , 2 3 = 𝐹 ( 𝑞 ) 2 𝜋 4 3 2 𝜋 4 3 2 3 𝜋
  • E 𝑤 𝜋 , 2 3 = 𝐹 ( 𝑞 ) 𝜋 2 3 𝜋 2 3 2 3 𝜋

Q14:

Suppose 𝑤 = 𝐹 ( 𝜙 ( 𝑥 , 𝑦 ) ) with 𝜙 = ( 𝜙 , 𝜙 , 𝜙 ) 1 2 3 and that 𝑞 = 𝜙 ( 𝑝 ) for a point 𝑝 2 . Express the gradient 𝑤 ( 𝑝 ) (viewed as a 1 × 2 matrix) in terms of the 1 × 3 matrix 𝐹 ( 𝑞 ) and a matrix of partial derivatives of 𝜙 .

  • A 𝑤 ( 𝑝 ) = 𝐹 ( 𝑞 ) 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 1 1 2 2 3 3
  • B 𝑤 ( 𝑝 ) = 𝐹 ( 𝑞 ) 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑦 1 2 3 1 2 3
  • C 𝑤 ( 𝑝 ) = 𝐹 ( 𝑞 ) 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑥 1 2 3 1 2 3
  • D 𝑤 ( 𝑝 ) = 𝐹 ( 𝑞 ) 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 1 1 2 2 3 3
  • E 𝑤 ( 𝑝 ) = 𝐹 ( 𝑞 ) 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 1 3 2 2 3 1

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.