 207
 14
 Problem Statement
 I want to check Stokes' theorem for the exercise below.
 Relevant Equations
 ##\int_{a} (\vec \nabla \times \vec F) \cdot d \vec a = \oint_{C} \vec F \cdot d \vec l##
I want to check Stokes' theorem for the following exercise:
Consider the vector field ##\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k##.
A closed curve ##C## lies in the plane ##x + y + z = 3##, oriented counterclockwise. The parametric representation of this curve is defined as:
$$C = (1+cost) \hat i + (1+sint) \hat j + (1  cost  sint) \hat k$$
Where ##t[0, 2\pi]##.
Stokes Theorem states:
$$\int_{a} (\vec \nabla \times \vec F) \cdot d \vec a = \oint_{C} \vec F \cdot d \vec l$$
Evaluating the LHS:
The curl of ##\vec F## :
$$\vec \nabla \times \vec F = 2x \hat k$$
This is an important point: How can I know the projection on the xy plane of the surface enclosed by C?
I am told that it is the circle with radius 1 and center (1, 1) but I do not know why. We can verify it graphically but I am seeking for a numerical method to get it, rather than having to draw the plot.
The unit normal vector to the circle is:
$$\hat n = (x1) \hat i + (y+1) \hat j$$
Setting up the integral:
$$d \vec a = \hat k dxdy$$
$$\int_{a} (\vec \nabla \times \vec F) \cdot d \vec a = \int_{a} 2x dxdy = 2\int_{0}^{2\pi}\int_{0}^{1} rcos \theta drd \theta = 0$$
BUT I GET 0... What does that imply?
Evaluating the RHS:
$$\oint_{C} \vec F \cdot d \vec l = \oint_{C}[ye^x dx + (x^2 + e^x)dy + z^2e^z dz]$$
But we know how the curve looks like:
$$x = 1 + cost$$
$$y = 1 + sint$$
$$z = 1  cost  sint$$
So it is a matter of solving the integral:
$$\oint_{C} = C_1 + C_2 + C_3$$
Where:
$$C_1 = \int_{0}^{2\pi} (1+sint) e^{1+cost}(sint)dt$$
$$C_2 = \int_{0}^{2\pi} [(1+cost)^2 + e^{1+cost}](cost)dt$$
$$C_3 = \int_{0}^{2\pi} (1costsint)^2e^{1costsint}(sintcost)dt$$
BUT How can I solve ##C_1##?
To sum up; these are the points I want to understand:
1) How to know the projection on the xy plane of the surface enclosed by C.
2) What does ##\int_{a} (\vec \nabla \times \vec F) \cdot d \vec a = \oint_{C} \vec F \cdot d \vec l = 0## mean? (##\vec F## is not a conservative field, so that is not the reason...).
3) How to solve ##C_1## kind of integrals (any trigonometric trick?). I tried to solve it using https://www.integralcalculator.com/ but founds no primitive...
Thanks.
Consider the vector field ##\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k##.
A closed curve ##C## lies in the plane ##x + y + z = 3##, oriented counterclockwise. The parametric representation of this curve is defined as:
$$C = (1+cost) \hat i + (1+sint) \hat j + (1  cost  sint) \hat k$$
Where ##t[0, 2\pi]##.
Stokes Theorem states:
$$\int_{a} (\vec \nabla \times \vec F) \cdot d \vec a = \oint_{C} \vec F \cdot d \vec l$$
Evaluating the LHS:
The curl of ##\vec F## :
$$\vec \nabla \times \vec F = 2x \hat k$$
This is an important point: How can I know the projection on the xy plane of the surface enclosed by C?
I am told that it is the circle with radius 1 and center (1, 1) but I do not know why. We can verify it graphically but I am seeking for a numerical method to get it, rather than having to draw the plot.
The unit normal vector to the circle is:
$$\hat n = (x1) \hat i + (y+1) \hat j$$
Setting up the integral:
$$d \vec a = \hat k dxdy$$
$$\int_{a} (\vec \nabla \times \vec F) \cdot d \vec a = \int_{a} 2x dxdy = 2\int_{0}^{2\pi}\int_{0}^{1} rcos \theta drd \theta = 0$$
BUT I GET 0... What does that imply?
Evaluating the RHS:
$$\oint_{C} \vec F \cdot d \vec l = \oint_{C}[ye^x dx + (x^2 + e^x)dy + z^2e^z dz]$$
But we know how the curve looks like:
$$x = 1 + cost$$
$$y = 1 + sint$$
$$z = 1  cost  sint$$
So it is a matter of solving the integral:
$$\oint_{C} = C_1 + C_2 + C_3$$
Where:
$$C_1 = \int_{0}^{2\pi} (1+sint) e^{1+cost}(sint)dt$$
$$C_2 = \int_{0}^{2\pi} [(1+cost)^2 + e^{1+cost}](cost)dt$$
$$C_3 = \int_{0}^{2\pi} (1costsint)^2e^{1costsint}(sintcost)dt$$
BUT How can I solve ##C_1##?
To sum up; these are the points I want to understand:
1) How to know the projection on the xy plane of the surface enclosed by C.
2) What does ##\int_{a} (\vec \nabla \times \vec F) \cdot d \vec a = \oint_{C} \vec F \cdot d \vec l = 0## mean? (##\vec F## is not a conservative field, so that is not the reason...).
3) How to solve ##C_1## kind of integrals (any trigonometric trick?). I tried to solve it using https://www.integralcalculator.com/ but founds no primitive...
Thanks.
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