Explanation Given: The given line integral is F = x 2 i + y 2 j + x 2 k and S Is the of the unit cube 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , 0 ≤ z ≤ 1 . Concept used: The divergence theorem: S is a smooth,orientable that encloses a soild region D in R 3 if F is a continuous vector fiels whose components have continuous partial derivatives in an open set containing D ,Then ∬ S F ⋅ N d s = ∬ D ∫ d i v ( F ) d V Here, N is the outward unit normal field for the S . Calculation: Consider the vector field, F = x 2 i + y 2 j + x 2 k And S is the of the unit cube 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , 0 ≤ z ≤ 1 . Since, ∬ S F ⋅ N d s = ∬ D ∫ d i v ( F ) d v ∬ S F ⋅ N d s = ∫ 0 1 ∫ 0 1 ∫ 0 1 d i v ( F ) d x d y d z d i v ( F ) Now,first find Then, d i v ( F ) = ∂ u ∂ x ( x , y , z ) + ∂ v ∂ y ( x , y , z ) + ∂ w ∂ z ( x , y , z ) Here, u = x 2 v = y 2 w = x 2 So, d i v

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Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 13, Problem 50SP

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To find:Thegiven line integral.

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Chapter 13, Problem 49SP

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Problem 1. Let F be the vector field (x²z, xyz, x²y), and S the paraboloid z = 1-x² – y², z 2 0 with upward pointing normal vector. 1. Calculate the curl of F, V × F. 2. Calculate (V x F) · dS. S

Incorrect. Use a computer or calculator with Euler's method to approximate the flow line through (1, 2) for the vector field v = y² i +1.1x² j using 5 steps with At = 0.1. Find the exact values of x1, ... , x5 and y1, ... , y5 and then fill in the blanks rounding your numbers to three decimal places. X1 = !Yı = i X2 i 1.2 , y2 3.273 X3 = i 1.3 Y3 = i 4.50283 X4 i 1.4 6.7162 X5 = i 1.5 Y5 = i 11.4427 eTextbook and Media Assistance Used Hint Assistance Used The vector field is given by v = y i + 1.1x² j , that is, the flow line (x (t), y (t)) satisfies x' (t) = y² y' (t) = 1.1x².

Find the curl and divergence for the given vector field. (Note that the order of the curl and divergence questions may vary in this quiz.) 1. Positive 2. Negative 3. 0 curl F 4. O div F 5. R 6. -K >

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Thegiven line integral.

Solution Summary: The author explains the divergence theorem: N is the outward unit normal field for the S, and F is a continuous vector fiels whose components have continuous partial derivatives.

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To find:Thegiven line integral.

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