Verify That Stokes' Theorem Is True for the Vector Field $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + \lambda \mathbf { j } + \mathbf { k }$

question 105

Short Answer

Verify that Stokes' Theorem is true for the vector field $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + \lambda \mathbf { j } + \mathbf { k }$ and the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 } , 0 \leq z \leq 1$ , oriented upward.

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Definitions:

Unforeseen Circumstances Rule

A principle in contract law that allows parties to a contract to be excused from fulfilling their obligations due to circumstances that were unforeseeable at the time the contract was made.

Preexisting Duty

An obligation that one party is already legally required to perform and which cannot serve as consideration for a new contract.

Bilateral Contract

A bilateral contract involves two parties who each promise to perform certain acts in exchange for the other's promise, establishing mutual obligations.

Illusory Promises

Promises that are vague or lack commitment, rendering them unenforceable in a contract because they fail to bind the promisor.