Consider F and C below. F(x, y, z) = (2xz + y^2) i + 2xy j + (x^2 + 6z^2) k C: x = t^2, y = t +...

Question:

Consider {eq}F{/eq} and {eq}C{/eq} below.

{eq}F(x, y, z) = (2xz + y^2) i + 2xy j + (x^2 + 6z^2) k {/eq}

C: {eq}x = t^2, y = t + 3, z = 3t - 1, 0 \leq t \leq 1{/eq}

(a) Find a function {eq}f{/eq} such that {eq}F = \nabla f{/eq}.

(b) Use part (a) to evaluate {eq}\int_C \nabla f \cdot dr{/eq} along the given curve {eq}C{/eq}.

Line Integral of the Vector Field:

Note that the line integral fo the vector field has the following formula {eq}\displaystyle \int F\cdot dr=\int_{a}^{b}F\left ( r(t) \right )\cdot r'(t)dt {/eq} where {eq}r(t) {/eq} is the vector equation of the curve and {eq}F {/eq} is the vector field.

Answer and Explanation: 1

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From the given above,

{eq}\displaystyle r(t)=\left \langle t^{2},t+3,3t-1 \right \rangle {/eq}

{eq}\displaystyle r'(t)=\left \langle 2t,1,3 \right...

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Vectors: Definition, Types & Examples

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Chapter 57 / Lesson 3
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Vectors describe amounts that extend in a direction and have a magnitude. Explore the definition, types, and examples of vectors and discover position vectors, unit vectors, and equal vs. parallel vectors.


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