problems_vector

# problems_vector - =(b z y y z x x y F ˆ 3 ˆ 3 2 ˆ 2 =(c...

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PHY2061 R. D. Field Problems Vector Analysis Page 1 of 1 Vector Analysis Problems Problem 1: Calculate the gradient of the scalar function 3 2 yz x f = . Problem 2: Let y x x y F ˆ ˆ + = ! . (a) Calculate r d F ! ! along Path 1 and Path 2 from point P 1 =(0,0) to P 2 =(1,1) as shown in the Figure. (b) Calculate the divergence , F ! ! , and the curl , F ! ! × . Problem 3: Calculate the curl and the divergence of each of the following vector functions. If the curl turns out to be zero, try to discover a scalar function, f(x,y,z), of which the vector function is the gradient. (a) z z y x x y x F ˆ 2 ˆ ˆ ) (
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Unformatted text preview: + = ! (b) z y y z x x y F ˆ 3 ˆ ) 3 2 ( ˆ 2 + + + = ! (c) z xz y x z x F ˆ 2 ˆ 2 ˆ ) ( 2 2 + + − = ! Problem 4: Prove that for any vector field, F ! , that the divergence of the curl of F ! is zero. ( i.e. Prove ) ( = × ∇ ⋅ ∇ F ! ! ! ). Problem 5: Let z z y y x x r ˆ ˆ ˆ + + = ! . Show that 3 = ⋅ ∇ r ! ! and = × ∇ r ! ! . Problem 6: Prove that F f F f F f ! ! ! ! ! ! × ∇ + × ∇ = × ∇ ) ( , where f is a scalar function and F ! is a vector field. y x Path 1 Path 2 (1,1) (0,0)...
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