Question

Determine whether the line integral of each vector field (in blue) along the oriented path (in red) is positive, negative, or zero

Answer 1

The answer in this question that determined the line integral of each vector field in blue along the semicircular, oriented path in red is a positive, negative or zero. 
Through the graph we will determined whether the line integral of each vector field is positive negative and zero. This is the answer in this question;
-Positive
-Negative
-Zero
-Negative
-Zero
-Zero

Answer 2

The line integral of each vector field (in blue) along the oriented path (in red) is positive, negative, or zero is shown in the picture

Explanation:

A line integral is the integral where the function to be integrated is evaluated along a curve.

For example, take a look at figure a which is the Line integral of a scalar field. If C is given by $x=x(t), y=y(t),t \epsilon[a , b]$ ,

then $\int\limits_ c {f(x,y)} \, ds = \int\limits^b_a {f(x(t),y(t))} \, ds$

where  $ds = \sqrt{\frac{dx}{dt} ^{2} + \frac{dy}{dt} ^{2} }$

As the area of the curved  vertical curtain along that curve under the  surface $z= f (x , y)$

The next example is figure b

Let $F(x,y,z) = M\^{i}+N\^{j}+P\^{k}$ where $M=M(x,y,z), N=N(x,y,z)$ and $P=P(x,y,z)$

Then $dr=dx\^{i}+dy\^{j}+dz\^{k}$ and $W= \int\limits^._C {F. \, dr$that calculates the work done in moving a particle over curve C with force F. This is shown in figure b.)a

Therefore the line integral of each vector field (in blue) along the oriented path (in red) is positive, negative, or zero can be determined as shown in figure below

Learn more about  the line integral brainly.com/question/13035260

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