Which of the following gives the relationship between line and surface integral?

Green’s Theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane D bounded by C. (See Figure 5.4.

How is a line integral converted into a surface integral?

Summary. Stokes’ theorem can be used to turn surface integrals through a vector field into line integrals. This only works if you can express the original vector field as the curl of some other vector field. Make sure the orientation of the surface’s boundary lines up with the orientation of the surface itself.

What is the difference between line surface and volume integral?

A line integral takes a vertical slice of area out of a 3d region. A surface integral takes the surface area of a 3d surface that we don’t have a formula for in basic geometry. A volume integral takes the volume of a 3d region.

Which theorem gives relation between surface integral and volume integral?

the divergence theorem
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.

What is the difference between line integral and double integral?

The line integral involves a vector field and the double integral involves derivatives (either div or curl, we will learn both) of the vector field. First we will give Green’s theorem in work form. The line integral in question is the work done by the vector field. The double integral uses the curl of the vector field.

Which theorem converts line integral into surface integral?

The Stoke’s theorem
3. Which of the following theorem convert line integral to surface integral? Explanation: The Stoke’s theorem is given by ∫A.

How does a line integral differ from the single variable integral?

A scalar line integral is defined just as a single-variable integral is defined, except that for a scalar line integral, the integrand is a function of more than one variable and the domain of integration is a curve in a plane or in space, as opposed to a curve on the x-axis.

Which of the following theorem gives the relation between surface integral and volume integral?

More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.

What is the difference between surface integrals and line integrals?

Line Integrals: A line integral is an integral where the function to be integrated is evaluated along a curve. Surface Integrals: A surface integral is a generalization of multiple integrals to integration over surfaces.

Why is the line integral of a vector equal to its curl?

It depends on the boundary of the surface. Hence, line integral of the vector is equal to the surface integral of the curl of the vector over the same boundary.

Why do we use U and V for surface integral?

Even in multidimensional space, a surface is a 2D entity and can be expressed completely and uniquely by two parameters, u and v (say) . This helps us to convert a surface integral into a double integral. Geometrically, it is akin to stretching, and flattening out a surface into a plane on u and v axes.

What is the difference between volume integral and multiple integral?

A volume integral is generalization of triple integral. A multiple integral is any type of integral. Let us go a little deeper. For simplicity, we will restrict our discussion to only Cartesian coordinates, but the same argument holds for other coordinates as well.