Ask Your Teacher   Use Stokes' Theorem to evaluate      C  F · dr  where C is oriented counterclockwise as viewed from above. F(x, y, z) = xyi + 2zj + 5yk,      C is the curve of intersection of the plane  x + z = 10  and the cylinder  x2 + y2 = 36.

Q: x² + y³. 2 = Consider the surface S of equation (1, 1, v), An equation for the tangent line tangent…

A: We have to find the equation of tangent line.

Q: Given w f(z,y, 2) = zy²z. Find: Ograd/(1,3,-2) 3. (b) equation of the tangent plane to the surface…

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Q: Find the angle between the surfaces x² + y² + z² = 41 and z = x² + y²-11 at the point (4, 0, 5):

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Q: Given the surface z = In(-2x² + 5y² – xy). Find the directional derivative of the function f at…

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Q: Indicate whether each of the following statements is True (T), or False (F). Explain your answers. 1…

A: See the solution in the given fig.1

Q: The surface z = x² - y² and the surface ryz = -30 intersect along a curve in R³. Find a vector…

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Q: Ř(u, v) = (u², v – 2u, u + v²)

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Q: Find a parametrization of the surface x³ + 15xy + z² : = 10 where x > 0 and use it to find the…

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Q: Find an equation of the tangent plane to the parametric surface

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Q: t-) For each surface below, find the normal vector and equation of the tangent plane at the given…

A: Use definition of tangent plane and norman vector.  According to guidelins, we solve only first 3…

Q: 2. Consider the vector-valued function 12t3 – 72t2 + 96t 15t – 60 6t – 24, if t > 0, t + 4, Ř(t = 4…

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Q: Find the equation of the tangent plane to the surface of the equation z = x2 + y2 - 2xy on the point…

A: Equation of tangent plane

Q: Consider the paraboloid given by the equation: z = x^2 + y^2 Find the equation of the tangent…

A: at point (1, 2, 5)at point (1, 2, 5)

Q: Let S be the surface of revolution of the curve C : z? + (y+ 5)² = 4. (a) Find a parametrization of…

A: Consider the curve C in zy-plane as shown: \ On rotating the curve around z-axis, we get The shape…

Q: Find the length of the parametric curve x = 2t + 1, y = 3 + 2tž, 0 < t <%.

A: Find the Length of the curve

Q: Find a parametrization of the curve of intersection of the surfaces z = x² – y² and z = x² + xy + 1.

A: The given curves are z=x2-y2 and z=x2+xy+1 To find the intersecting curve : Let us equate both the…

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: