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Hudman57
24.06.2020 •
Mathematics
Consider F and C below. F(x, y, z) = y2 sin(z) i + 2xy sin(z) j + xy2 cos(z) k C: r(t) = t2 i + sin(t) j + t k, 0 ≤ t ≤ π (a) Find a function f such that F = ∇f. f(x, y, z) = (b) Use part (a) to evaluate C ∇f · dr along the given curve C.
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Ответ:
a)![f (x,y,z)= xy^2\sin(z)](/tpl/images/0693/4958/49556.png)
b)![\int_C F \cdot dr =0](/tpl/images/0693/4958/57d79.png)
Step-by-step explanation:
Recall that given a function f(x,y,z) then
. To find f, we will assume it exists and then we will find its form by integration.
First assume that F =
. This implies that
This must be equal to the second component of F. Then
This implies that
, which means that g depends on z only. So ![f(x,y,z) = xy^2\sin(z) + g(z)](/tpl/images/0693/4958/b0198.png)
Taking the derivative with respect to z and making it equal to the third component of F, we get
which implies that
which means that g(z) = K, where K is a constant. So
b) To evaluate
we can evaluate it by using f. We can calculate the value of f at the initial and final point of C and the subtract them as follows.
Recall that
so ![f(r(\pi)) = \pi^2\cdot 0 \cdot \sin(\pi) = 0](/tpl/images/0693/4958/639e1.png)
Also
so ![f(r(0)) = 0^2\cdot 0 \cdot \sin(0) = 0](/tpl/images/0693/4958/1ce41.png)
So![\int_C F \cdot dr =0](/tpl/images/0693/4958/57d79.png)
Ответ:
Step-by-step explanation:
correct Question :A rectangle has a side length and wide of 9 centimeters and 6 centimeters. What is the area of the rectangle Given that :A rectangle has a side length and wide of 9 centimeters and 6 centimeters. to find :area of rectangle. formulas used :A = l × wwhere,
A = area of rectangle l = lenth of rectangle W = wide of rectangle explanation :⟼ l = 9 cm
⟼ W = 6 cm
⟼ Area = l × b
⟼ Area = 9 × 6 cm
⟼ Area = 54 cm².
∴ the area of rectangle is 54 cm².