Worksheet: Conservative Vector Fields

In this worksheet, we will practice determining whether a vector field is conservative or not and finding the potential function of conservative vector fields.

Q1:

Is there a potential 𝐹 ( 𝑥 , 𝑦 ) for f i j ( 𝑥 , 𝑦 ) = ( 8 𝑥 𝑦 + 3 ) + 4 𝑥 + 𝑦 ? If so, find one.

  • A yes, 𝐹 ( 𝑥 , 𝑦 ) = 4 𝑥 𝑦 + 2 𝑦 + 3 𝑥
  • B yes, 𝐹 ( 𝑥 , 𝑦 ) = 4 𝑥 𝑦 1 2 𝑦 + 3 𝑥
  • C no
  • D yes, 𝐹 ( 𝑥 , 𝑦 ) = 4 𝑥 𝑦 + 1 2 𝑦 + 3 𝑥
  • E yes, 𝐹 ( 𝑥 , 𝑦 ) = 4 𝑥 𝑦 2 𝑦 + 3 𝑥

Q2:

Is there a potential 𝐹 ( 𝑥 , 𝑦 ) for f i j ( 𝑥 , 𝑦 ) = 𝑥 𝑦 ? If so, find one.

  • A yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦
  • B yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦
  • C yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 2 + 𝑦 2
  • D no
  • E yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 2 𝑦 2

Q3:

Is there a potential 𝐹 ( 𝑥 , 𝑦 ) for f i j ( 𝑥 , 𝑦 ) = 𝑦 + 3 𝑥 + 2 𝑥 𝑦 ? If so, find one.

  • A yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 + 𝑥
  • B yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 𝑥
  • C yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 + 𝑥
  • D yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 + 𝑦 𝑥
  • E no

Q4:

Is there a potential 𝐹 ( 𝑥 , 𝑦 ) for 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑥 𝑦 + 2 𝑥 𝑥 𝑦 + 𝑥 𝑦 c o s s i n i j ? If so, find one.

  • A yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑥 𝑦 2 𝑥 𝑦 s i n c o s
  • B no

Q5:

Is there a potential 𝐹 ( 𝑥 , 𝑦 ) for f i j ( 𝑥 , 𝑦 ) = 𝑥 𝑦 𝑥 𝑦 ? If so, find one.

  • A yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦
  • B yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦
  • C no
  • D yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 2 + 𝑥 𝑦 2
  • E yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 2 𝑥 𝑦 2

Q6:

State whether or not the vector field f i j k ( 𝑥 , 𝑦 , 𝑧 ) = 𝑎 + 𝑏 + 𝑐 where 𝑎 , 𝑏 , 𝑐 are constants has a potential in .

  • Ano
  • Byes

Q7:

State whether or not the vector field f i j k ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 𝑦 𝑥 𝑦 𝑧 + 𝑦 𝑧 has a potential in .

  • Ayes
  • Bno

Q8:

Is there a potential 𝐹 ( 𝑥 , 𝑦 ) for f i j ( 𝑥 , 𝑦 ) = 𝑦 𝑥 ? If so, find one.

  • Ano
  • B yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑦 2 𝑥 2 + 𝐾
  • C yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 + 𝐾
  • D yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑦 2 𝑥 2
  • E yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦

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