Find that F = 12x²yi + (4x + 8yz²)j + 8y² zk is conservative, and find a function f such that F = Vf, and use the result to evaluate F • dr, where C is any curve from A = (0, 0,0) to B = (-3,2, 3).

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A: Linear approximation

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A: Proceed as shown below

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Q: 6. (a) Show that F = (3a²y+2, x3 + 3) is conservative. (b) Evaluate Sc(3x²y +2)dx + (x³ + 3)dy where…

A: According to the problem, we have

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A: This question is related to differential equations, we will solve it using given information.

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Q: If F(t) = 2xyi + xyz2j + ex+yk, then what is the divergence of F? Select one: A. 2yi + xz2j + ex+yk…

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A: Answer: In oerder to use gradients we introduce a new variable c= 7x5+cos(5y)+ln(9z) Therefore the…

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: