State two characteristics of line integrals of scalar field

Expert Solution

milcah answered 2 years ago

State and prove the Weighted Mean Value Theorem for integrals.

The projection or scalar component of a force along a line is found by: A. Finding...

The projection or scalar component of a force along a line is found by: A. Finding the direction cosines for the force with respect to a rectangular coordinate system. B. Taking the cross product of the force and the vector representation of the line. C. Taking the scalar product of the force and the vector representation of the line. D. Finding the direction cosines for the line with respect to a rectangular coordinate system.

a) show that the vector field F= <siny, xcosy, -sinz> is conservative by finding a scalar...

a) show that the vector field F= <siny, xcosy, -sinz> is conservative by finding a scalar f such that ∇f=F. b) use this fact to ecaluate ∫C F•dr along the given curve C where C is the line segment from (0,0,0) to (4, π/2, π/2).

(a) State Newton’s law of gravity in scalar form and vector form. (b) State how potential...

(a) State Newton’s law of gravity in scalar form and vector form. (b) State how potential energy and forces in 1D are related and use this to determine an expression for the gravitational potential energy, choosing an appropriate zero-point. (c) A thin, 1D homogeneous rod of material has a mass M and length L. The rod is bent into a semi-circle. Determine an expression for the strength of the gravitational ﬁeld at the centre of curvature of the circular arc.

a) ln(x2 + y2) calculate the Laplacian ∇2 of the scalar field using both cartesian and...

a) ln(x2 + y2) calculate the Laplacian ∇2 of the scalar field using both cartesian and cylindrical coordinate systems b) (x2 + y2 + z2)-1/2 calculate the Laplacian ∇2 of the scalar field using both cartesian and spherical coordinate systems

QUESTION 1: Are electric force and electric field vector or scalar quantities? electric force is a...

QUESTION 1: Are electric force and electric field vector or scalar quantities? electric force is a scalar electric field is a vector electric force is a vector electric field is a scalar electric force and electric field are both scalars electric force and electric field are both vectors QUESTION 2: When an electron is placed in an electric field it experiences an electrostatic force towards the North. In what direction is the electric field? Towards the East Towards the West Towards...

Describe the magnetic field line of force of iron fling 2. Describe the magnetic field line...

Describe the magnetic field line of force of iron fling 2. Describe the magnetic field line of force of a needle

Hello, I am a bit confused as how line integrals can exist in 2D. For 2D,...

Hello, I am a bit confused as how line integrals can exist in 2D. For 2D, we are dealing with two variables: x and y. For 3D, we are dealing with three variables: x,y and z=f(x,y), such that f(x,y) = x+y. Since line integrals use the x and y components as inputs and f(x,y) = z as the output, should not line integrals exist in 3D always?

With regard to multi variable calculus can some explain line and surface integrals in detail. Thank...

With regard to multi variable calculus can some explain line and surface integrals in detail. Thank you. Provide an example problem and solve too thanks!

Fundamental Theorem of Line Integrals Consider the line integral: ∮_C▒〈6xy+6x, 3x^2-3 cos⁡(y) 〉 ∙dr ⃗ where...

Fundamental Theorem of Line Integrals Consider the line integral: ∮_C▒〈6xy+6x, 3x^2-3 cos⁡(y) 〉 ∙dr ⃗ where C is the line segment from the origin to (4, 5). Evaluate this integral by rewriting the integral as a standard single variable integral of the parameter t. Without finding a potential function show that the integrand is a gradient field. Find a potential function for the vector field. Use the Fundamental Theorem of Line Integrals to evaluate this integral.

Question