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Chapter 2: Q35E (page 65)

Using condition (5), show that the right-hand side of (10) is independent of x by showing that its partial derivative with respect to x is zero. [Hint: Since the partial derivatives of M are continuous, Leibniz鈥檚 theorem allows you to interchange the operations of integration and differentiation.]

Short Answer

Expert verified

Leibniz theorem allows interchanging the operations of integration and Differentiation.

Step by step solution

01

Show that RHS of equation (10) is independent of x

The condition of exactness is 鈭�M鈭�y=鈭�N鈭�x.

Now, take equation (10) and partially differentiate the RHS w.r.t. x

鈭�鈭�x(RHS)=鈭�鈭�xN(x,y)-鈭�鈭�x鈭�鈭�y鈭�xoxM(t,y)dt=鈭�N(x,y)鈭�x-鈭�鈭�y鈭�鈭�x鈭�xoxM(t,y)dt=鈭�N(x,y)鈭�x-鈭�鈭�y鈭�xox鈭�鈭�xM(t,y)dt

02

Apply Leibniz rule

Apply the Leibniz rule, then

鈭�鈭�x(RHS)=鈭�N(x,y)鈭�x-鈭�鈭�yM(x,y)

Now, uses the condition (5) then

鈭�鈭�xN(x,y)-鈭�鈭�x鈭�鈭�y鈭�xoxM(t,y)dt=0

Therefore the RHS of the equation (10) is independent of x.

Hence, that the right-hand side of (10) is independent of x

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