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hw1 - 0 →(0 1 0 →(1 1 0(b the path from(0 0 →(1 0...

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Electricity and Magnetism I (P331) Homework 1 Due: Wednesday, September 9, 4:00 PM 1. Determine the angle between the diagonal of a cube and an adjacent edge. Hint: use the unit cube – the diagonal is given by ˆ x + ˆ y + ˆ z . 2. (a) Prove that | A × B | is equal to the area of the parallelogram defined by A and B . (b) Show that A · ( B × C ) is equal to the volume of the parallelepiped defined by A , B , and C . From this result prove that A · ( B × C ) = ( A × B ) · C . 3. Prove the following identities: (a) ∇ · ( g F ) = g ∇ · F + g · F (b) ∇ × ( g F ) = g ∇ × F + g × F (c) ∇ · ( F × G ) = G · ( ∇ × F ) - F · ( ∇ × G ) (d) ∇ × ( ∇ × F ) = ( ∇ · F ) - ∇ 2 F 4. (a) Prove that the divergence of a curl is always zero, and check it for the function v = xz 2 ˆ x + 2 xy ˆ y + zy ˆ z . (b) Prove that the curl of a gradient is always zero, and check it for the function f = x 3 + zx + y 2 z . 5. Consider the function g = x 3 + 3 xy 2 . Check the theorem for gradients by showing that the integral of g from (0 , 0 , 0) to (1 , 1 , 0) is the same for the following three paths: (a) the path from (0
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Unformatted text preview: , , 0) → (0 , 1 , 0) → (1 , 1 , 0) (b) the path from (0 , , 0) → (1 , , 0) → (1 , 1 , 0) (c) the path from (0 , , 0) → (1 , 1 , 0) along the parabola y = x 2 6. (a) Sketch a picture of the vector ﬁeld F = x ˆ x . (b) Calculate directly the ﬂux of F outward through the surface of the unit cube deﬁned by ≤ x, y, z ≤ 1. (c) Calculate the ﬂux using the divergence theorem. 7. For the following problem work in rectangular coordinates . (a) Sketch a picture of the vector ﬁeld F = ˆ z × x ˆ x (b) Calculate directly the line integral of F around a circle in the xy plane, centered at 0 with a radius a . (c) Calculate the line integral by using Stokes’ theorem. 8. Rewrite F from the problem above in cylindrical coordinates and rework the problem in cylindrical coordinates. 1...
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