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(π‘₯,𝑦) for fij(π‘₯,𝑦)=(8π‘₯𝑦+3)+4ο€Ήπ‘₯+π‘¦ο…οŠ¨? If so, find one.

  • Ayes, 𝐹(π‘₯,𝑦)=4π‘₯π‘¦βˆ’2𝑦+3π‘₯
  • Byes, 𝐹(π‘₯,𝑦)=4π‘₯𝑦+2𝑦+3π‘₯
  • Cyes, 𝐹(π‘₯,𝑦)=4π‘₯𝑦+12𝑦+3π‘₯
  • Dyes, 𝐹(π‘₯,𝑦)=4π‘₯π‘¦βˆ’12𝑦+3π‘₯
  • Eno

Q2:

Is there a potential 𝐹(π‘₯,𝑦) for fij(π‘₯,𝑦)=π‘₯βˆ’π‘¦? If so, find one.

  • Ayes, 𝐹(π‘₯,𝑦)=π‘₯2+𝑦2
  • Byes, 𝐹(π‘₯,𝑦)=π‘₯+π‘¦οŠ¨οŠ¨
  • Cno
  • Dyes, 𝐹(π‘₯,𝑦)=π‘₯βˆ’π‘¦οŠ¨οŠ¨
  • Eyes, 𝐹(π‘₯,𝑦)=π‘₯2βˆ’π‘¦2

Q3:

Is there a potential 𝐹(π‘₯,𝑦) for fij(π‘₯,𝑦)=𝑦+3π‘₯+2π‘₯π‘¦οŠ¨οŠ¨? If so, find one.

  • Ayes, 𝐹(π‘₯,𝑦)=π‘₯𝑦+𝑦π‘₯
  • Byes, 𝐹(π‘₯,𝑦)=π‘₯𝑦+π‘₯
  • Cyes, 𝐹(π‘₯,𝑦)=π‘₯π‘¦βˆ’π‘₯
  • Dyes, 𝐹(π‘₯,𝑦)=π‘₯𝑦+π‘₯
  • Eno

Q4:

Is there a potential 𝐹(π‘₯,𝑦) for 𝑓(π‘₯,𝑦)=ο€Ήπ‘₯π‘₯𝑦+2π‘₯π‘₯𝑦+π‘₯π‘¦οŠ©οŠ¨cossinij? If so, find one.

  • Ayes, 𝐹(π‘₯,𝑦)=π‘₯π‘₯π‘¦βˆ’2π‘₯π‘¦οŠ¨sincos
  • Bno

Q5:

Is there a potential 𝐹(π‘₯,𝑦) for fij(π‘₯,𝑦)=π‘₯π‘¦βˆ’π‘₯π‘¦οŠ¨οŠ©? If so, find one.

  • Ayes, 𝐹(π‘₯,𝑦)=π‘₯π‘¦οŠ¨οŠ¨
  • Byes, 𝐹(π‘₯,𝑦)=βˆ’π‘₯π‘¦οŠ¨οŠ¨
  • Cno
  • Dyes, 𝐹(π‘₯,𝑦)=π‘₯𝑦2+π‘₯𝑦2
  • Eyes, 𝐹(π‘₯,𝑦)=π‘₯𝑦2βˆ’π‘₯𝑦2

Q6:

State whether or not the vector field fijk(π‘₯,𝑦,𝑧)=π‘Ž+𝑏+𝑐 where π‘Ž, 𝑏, 𝑐 are constants has a potential in β„οŠ©.

  • Ayes
  • Bno

Q7:

State whether or not the vector field fijk(π‘₯,𝑦,𝑧)=π‘₯π‘¦βˆ’ο€Ήπ‘₯βˆ’π‘¦π‘§ο…+π‘¦π‘§οŠ¨οŠ¨ has a potential in β„οŠ©.

  • Ano
  • Byes

Q8:

Is there a potential 𝐹(π‘₯,𝑦) for fij(π‘₯,𝑦)=π‘¦βˆ’π‘₯? If so, find one.

  • Ayes, 𝐹(π‘₯,𝑦)=𝑦2βˆ’π‘₯2+𝐾
  • Byes, 𝐹(π‘₯,𝑦)=π‘₯𝑦
  • Cyes, 𝐹(π‘₯,𝑦)=π‘₯𝑦+𝐾
  • Dno
  • Eyes, 𝐹(π‘₯,𝑦)=𝑦2βˆ’π‘₯2

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