Final answer:
The mixed partial derivatives confirm that the given differential form is exact, allowing us to find a potential function and evaluate the integral using the endpoints.
Step-by-step explanation:
To show that the given form under the integral sign is exact in space, we need to ensure that mixed partial derivatives of some potential function are equal. The given differential form is ze^zx dx + dy + xe^xz dz. Let's call this the differential of a potential function, Φ, such that dΦ = ze^zx dx + dy + xe^xz dz. If this is true, then the function must satisfy:
- (F3)_y = (F2)_z
- (F1)_z = (F3)_x
- (F1)_y = (F2)_x
Where F1, F2, and F3 are the components of the differential form corresponding to dx, dy, and dz respectively. In our case, F1 = ze^zx, F2 = 1, and F3 = xe^xz. Checking the partial derivatives, we find:
- (F3)_y = 0 since there is no y-dependence in F3.
- (F2)_z = 0 since F2 is a constant.
- (F1)_z = e^zx + zxe^zx
- (F3)_x = e^zx + zxe^zx
Since (F1)_z = (F3)_x and (F3)_y = (F2)_z, the form is exact. We can now evaluate the integral, which is done simply by finding the potential function Φ where dΦ gives our differential form, and taking the difference Φ(0,8,7) - Φ(2,3,0).