Let C be a simple closed smooth curve that lies in the plane x + y + z = 1 .Show that the line integral ∫ C z d x − 2 x d y + 3 y d z depends only on the area of the region enclosed by C and not on the shape of C or its location in the plane.
Let C be a simple closed smooth curve that lies in the plane x + y + z = 1 .Show that the line integral ∫ C z d x − 2 x d y + 3 y d z depends only on the area of the region enclosed by C and not on the shape of C or its location in the plane.
Solution Summary: The author explains that the given line integral depends only on the area of the region enclosed by C and not on its shape or location in the plane.
Let C be a simple closed smooth curve that lies in the plane
x
+
y
+
z
=
1
.Show that the line integral
∫
C
z
d
x
−
2
x
d
y
+
3
y
d
z
depends only on the area of the region enclosed by C and not on the shape of C or its location in the plane.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.