These problems are done with the CAS. See Hand Solutions for details.

The vector field F is not conservative since it does not possess a potential function f; i.e., a function such that F, the gradient of f. In the code below, we use MATLAB's potential function to show this.

By hand, note that with F we do not have for these partial derivatives, which are continuous on the xy-plane. Thus by Theorem 5 on page 1090 of the textbook, F is not conservative.

syms x y real

P = x*y + y^2, Q = x^2 + 2*x*y

F = [P Q]

f = potential(F, [x y]) % NaN or Not-a-Number means DNE (does not exist) in this context.

dP_dy = diff(P,y), dQ_dx = diff(Q,x)

The vector field F is conservative since it has a potential function f; i.e., a function such that F, the gradient of f.

(Again, see Hand Solutions for a check that this is true and how to actually construct the potential function by hand.)

syms x y

F = [y*exp(x) exp(x)+exp(y)]

f = expand( potential(F, [x y]) )

syms x y

F = [2*x*y + y^(-2) x^2 - 2*x*y^(-3)]

f = expand( potential(F, [x y]) ) % potential function f

syms x y

F = [log(y) + y/x log(x) + x/y]

f = expand( potential(F, [x y]) ) % potential function f

The path C is the arc of the hyperbola from to . The line integral of F along this path may be computed by the Fundamental Theorem for Line Integrals (FTLI) once we find a potential function f.

syms x y

F = [3 + 2*x*y^2 2*x^2*y]

% (a)

f(x,y) = expand( potential(F, [x y]) ) % potential function f

% (b)

I = f(4,1/4) - f(1,1)

The path C is the arc of the ellipse r, from to . The line integral of F along this path may be computed by the Fundamental Theorem for Line Integrals (FTLI) once we find a potential function f.

syms x y

F = exp(x*y) * [1+x*y x^2]

% (a)

f(x,y) = expand( potential(F, [x y]) ) % potential function f

% (b)

I = f(0,2) - f(1,0)

The path C is the space curve r, from to . The line integral of F along this path may be computed by the Fundamental Theorem for Line Integrals (FTLI) once we find a potential function f.

syms x y z

F = [y^2*z + 2*x*z^2 2*x*y*z x*y^2 + 2*x^2*z]

% (a)

f(x,y,z) = expand( potential(F, [x y z]) ) % potential function f

% (b)

I = f(1,2,1) - f(0,1,0)

The path C is the space curve r, from to . The line integral of F along this path may be computed by the Fundamental Theorem for Line Integrals (FTLI) once we find a potential function f.

syms x y z

F = [sin(y) x*cos(y) + cos(z) -y*sin(z)]

% (a)

f(x,y,z) = expand( potential(F, [x y z]) ) % potential function f

% (b)

I = f(1,pi/2,pi) - f(0,0,0)

I_appx = double(I)

I_appx = -0.5708

Here C is any path in the plane from to . The line integral of F along this path may be computed by the Fundamental Theorem for Line Integrals (FTLI) once we find a potential function f.

syms x y

F = [sin(y) x*cos(y) - sin(y)]

% (a)

f(x,y) = expand( potential(F, [x y]) ) % potential function f

% (b)

I = f(1,pi) - f(2,0)

Work is computed via the FTLI, analogous to #9.

syms x y

F = [2*x+y x]

% (a)

f(x,y) = expand( potential(F, [x y]) ) % potential function f

% (b)

work = f(4,3) - f(1,1)