Stokes' Theorem is a fundamental theorem in vector calculus which relates a surface integral of a vector field over a given surface S to a line integral of the same vector field along the boundary curve of that surface. In mathematical form, Stokes' Theorem can be expressed as:
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} where,
- F is a Vector Field
- ∇×F is Curl of F
- dS is Vector Area Element of Surface
- dr is Line Element along Boundary Curve
In this article, we'll see how to apply Stokes' Theorem effectively. Here are some Stokes Theorem Practice Problems to help you master the application of Stokes' Theorem
What is Stokes' Theorem?
Stokes' Theorem plays a major role in connecting the surface integrals of a vector field's curl with line integrals along the boundary of the surface. This theorem is used and is a very useful tool in various fields like fluid dynamics, electromagnetism, and more, helping us understand how tiny details of a field can add up to big effects on the surface. It's like connecting the dots between what's going on inside a surface and what we can see on the outside.
Stokes Theorem Practice Problems - Solved
Here are some Stokes Theorem Practice Problems designed to help you understand and apply Stokes' Theorem:
Problem 1: Given the vector field F = (y, -x, z2 ) and the surface S which is the upper half of the unit sphere x2 +y2 +z2 = 1 (with z ≥ 0), verify Stokes' Theorem by calculating both the surface integral and the line integral.
∇×F = (0, 0, -2)
\iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = -2\iint_{S} \sin \theta \, d\theta \, d\phi = -4\pi \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \oint_{\partial S} (\sin \phi, -\cos \phi, 0) \cdot (-\sin \phi \, d\phi, \cos \phi \, d\phi, 0) = \oint_{\partial S} \cos(2\phi) \, d\phi = 0 Stokes' Theorem not verifed.
Problem 2: Consider the vector field F = ( x, y, z) and the surface S which is the part of the plane x + y + z = 1 in the first octant, bounded by the coordinate planes. Verify Stokes' Theorem.
∇×F = (0, 0, 0)
\iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_{S} (0, 0, 0) \cdot (1, 1, 1) \, du \, dv = 0
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \int_{0}^{1} x \, dx + \int_{0}^{1} y \, dy + \int_{0}^{1} (2x - 1) \, dx = 0 + 0 + 0 = 0 Stokes' Theorem holds as 0 = 0.
Problem 3: For the vector field F = (y2, x2, z) and the surface S which is the part of the cylinder x2 + y2 = 1 for 0 ≤ z ≤ 2, verify Stokes' Theorem.
∇×F = (0, 0, 2xy)
\iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_{S} 2xy \, dA = \int_{0}^{2\pi} \int_{0}^{2} 2\cos\theta \sin\theta \, dz \, d\theta = 0
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \oint_{\partial S} (y^2, x^2, 2) \cdot (dx, dy, dz) = 0 Stokes' Theorem holds as 0 = 0.
Problem 4: Given the vector field F=(yz,xz,xy) and the surface S which is the part of the sphere x2+y2+z2 =9 in the first octant, verify Stokes' Theorem.
∇×F = (z-y, x-z, y-x)
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} (3\cos\theta - 3\sin\phi \sin\theta) \cdot (27 \sin^2 \theta \, d\phi \, d\theta) \\= \frac{81\pi}{2}
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \int_0^{\frac{\pi}{2}} (9 \sin\phi \cos\phi, 9 \cos^2\phi, 0) \cdot (-3 \sin\phi \, d\phi, 3 \cos\phi \, d\phi, 0) \\= \frac{81\pi}{2} Stokes' Theorem holds as 81?/2 = 81?/2.
Problem 5: Consider the vector field F = (x2, y2, z2) and the surface S which is the part of the plane x + 2y + 3z = 6 in the first octant, verify Stokes' Theorem.
∇×F = (0, 0, 0)
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S (0, 0, 0) \cdot \mathbf{n} \, dA \\= 0
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = 0 Stokes' Theorem holds as 0 = 0.
Problem 6: For the vector field F = (x3, y3, z3) and the surface S which is the part of the cylinder x2 + y2 = 4 for 0 ≤ z ≤ 3, verify Stokes' Theorem.
∇×F = (0, 0, 0)
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S (0, 0, 0) \cdot \mathbf{n} \, dA \\= 0
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = 0 Stokes' Theorem holds as 0 = 0.
Problem 7: Given the vector field F = (y, z, x) and the surface S which is the upper half of the sphere x2 + y2 + z2 = 1, verify Stokes' Theorem.
∇×F = (1 ,1 ,1)
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S 1 \, dS \\ = 2\pi
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \oint_{\partial S} (-\sin\phi \, d\phi, \cos\phi \, d\phi, 0) \\ = 2\pi Stokes' Theorem holds as 2? = 2?
Problem 8: Consider the vector field F=(z,y,x) and the surface S which is the part of the plane x+y+z=2 in the first octant, verify Stokes' Theorem.
∇×F = (1, -1, 0)
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S (1, -1, 0) \cdot (-1, -1, 1) \, dx \, dy = \iint_S (1 + 1) \, dx \, dy = 2 \int_0^2 \int_0^{2-x} \, dy \, dx = 2 \left[ \int_0^2 (2 - x) \, dx \right] = 2 \left[ 2x - \frac{x^2}{2} \right]_0^2 = 2(2) = 4
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} &= \int_0^2 (0, 0, t) \cdot (1, 0, -1) \, dt + \int_0^2 (2-t, t, 0) \cdot (-1, 1, 0) \, dt + \int_0^2 (2, 0, 2-t) \cdot (0, -1, 1) \, dt \\ &= 0 + 2 + 2 \\ &= 4 Stokes' Theorem holds as 4 = 4.
Stokes Theorem Practice Problems - Solved
Q1: Given the vector field F = (z, x, y) and the surface S which is the upper half of the sphere x2 + y2 + z2 = 4, verify Stokes' Theorem.
Q2: For the vector field F = (y2, x2, z) and the surface S which is the part of the plane x + 2y + z = 6 in the first octant, bounded by the coordinate planes, verify Stokes' Theorem.
Q3: Given the vector field F = (yz, xz, xy) and the surface S which is the part of the sphere x2 + y2 + z2 = 1 in the first octant, verify Stokes' Theorem.
Q4: For the vector field F = (x2, y2, z2) and the surface S which is the part of the plane x + y + z = 1 in the first octant, verify Stokes' Theorem.
Q5: Consider the vector field F = (x3, y3, z3) and the surface S which is the part of the cylinder x2 + y2 = 4 for 0 ≤ z ≤ 2, verify Stokes' Theorem.
Q6: Given the vector field F = (z, y, x) and the surface S which is the part of the sphere x2 + y2 + z2 = 9 in the first octant, verify Stokes' Theorem.
Q7: For the vector field F = (x2, y2, z2) and the surface S which is the part of the plane x + 2y + 3z = 6 in the first octant, verify Stokes' Theorem.
Q8: Given the vector field F = (x, y, z) and the surface S which is the part of the cylinder x2 + y2 = 1 for 0 ≤ z ≤ 3, verify Stokes' Theorem.
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