Chapter 1: Q1.34P (page 36)
Test Stokes' theorem for the function , using the triangular shaded area of Fig. 1.34.
Short Answer
The left and right side gives same result. Hence, strokes theorem is verified.
Chapter 1: Q1.34P (page 36)
Test Stokes' theorem for the function , using the triangular shaded area of Fig. 1.34.
The left and right side gives same result. Hence, strokes theorem is verified.
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Get started for freeProve product rules (i), (iv), and (v)
(a) Find the divergence of the function
(b) Find the curlof .Test your conclusion using Prob. 1.61b. [Answer:]
Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that:
(a). [Hint:Let v = cT, where c is a constant, in the divergence theorem; use the product rules.]
(b). [Hint:Replace v by (v x c) in the divergence
theorem.]
(c) . [Hint:Let in the
divergence theorem.]
(d). [Comment:This is sometimes
called Green's second identity; it follows from (c), which is known as
Green's identity.]
(e) [Hint:Let v = cT in Stokes' theorem.]
Compute the line integral of
around the path shown in Fig. 1.50 (the points are labeled by their Cartesian coordinates).Do it either in cylindrical or in spherical coordinates. Check your answer, using Stokes' theorem. [Answer:3rr /2]
Calculate the volume integral of the function over the tetrahedron with comers at (0,0,0), (1,0,0), (0,1,0), and (0,0,1).
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