b) Show that F - dr is independent of its path, given that y F(r, y, 2) = 1+. + tan-2) i+ tanrj+ k. Hence, find the potential function of F and then evaluate the work done by F moving an object along the smooth curve C, the line segment from (1,0, –1) to (0,0, 0) then along the parabola z = y,r = 0 to (0, –1, 1).

Q: 4. (a) Determine the directional derivative of f(x, y, z) = zety at the point (1, −1, 1) in the…

A:

Q: An equation for the tangent plane to the surface : = (xty at the point P(1, 1, 4) is a) 8x + 8y -: =…

A: Equation of a tangent to the surface z=f(x,y) at point (a,b,f(a,b)) is given by…

Q: B- Find the directional derivative of the function W = x² + xy + z³ at the point P: (2,1,1) in the…

A:

Q: 1. Consider a point particle with position vector r = (x, y, z) in Cartesian coordinates, moving…

A: Given that position vector of a particle at any instant t, so But given thatso we get (d) Now from…

Q: 6. (a) Find the directional derivative of w = x²y² at the point (1, -3) in the direc- 5T 5T tion of…

A:

Q: Q2) Find the vectors the curve r(t) = at the point (11) of

A: First of all find r'(t) and the length of r'(t). Here, r'(t)=<2t,2t2,1> |r'(t)|=4t2+4t4+1…

Q: One of the following vector functions is a gradient Select one: O (xy? + 2y?)7+ (2y³ – x²y+ 2x)j O…

A:

Q: Solution r(t) = 6 cos(t) i + 6 sin(t)j + 3tk. The parametric equations for this curve are x= y = 6…

A:

Q: 4. Determine the directional derivatives of $(x, y) = tan"2+3 sin ). at the point (1,1) in the…

A:

Q: Use the product rule to find the derivative of the following function, then select the correct…

A:

Q: TRUE OR FALSE. A plane curve C defined by the parametric equations x = f(t) and y = g(t), where t is…

A: check the given statement is true or not.

Q: 6. Given f(x, y) = 2x³y=x²y³, a) find the directional derivative of f(x, y) at (1, 1) in the…

A:

Q: determine wave direction in the function f(x,y) x^y — x²y³ decreases faster at the point (2,-3)

A: Wave direction in the function f(x,y)= x4y-x2y3 that decreases faster at the point (2,-3) is…

Q: 10. Given the curve C defined by the vector-valued function r(t) = a) find an equation of the line…

A:

Q: Find r'(t), r(t), and r'(to) for the given value of to. r(t) = 3 cos(t)i + 3 sin(t)j, to r'(t) =…

A: The given vector valued function

Q: 18.) Compute the directional derivative of f in the given ✓ given points P. 3 a) f(x, y, z) = xy² +…

A:

Q: Compute the directional derivative of f(r. y) = = at the point (1, 1) in a irection normmal to the…

A:

Q: Find the equation of the tangent line to right branch of the hyperbola x = sec t, y tant, -sts at…

A: Given query is to find the equation of tangent line.

Q: Suppose F(x, y) = x² + y²j and C is the line segment segment from point P = (-3,-4) to Q = (-3, 1).…

A:

Q: 3xyz = 63 and 3x² + 4y² + 8z² = 423 at point (−3, 1, −7). ř(t) = with -∞ < t <∞

A: The parametric form of a line that passes through x0, y0, z0 and has the direction ratios: a, b, c…

Q: 124 + 2452 + 35a3 + 24x4 = 24 356 + 24x2 + 35x3 + 754 = 2 241 + 56x2 + 35x3 + 67x4 = 47 256 + 87x2 +…

A:

Q: Choose the direction vector for which the function f defined by f(x,v) = sin(7x)cos(13y) has the…

A:

Q: a) Compute the gradient of the function f(x, y, z) = e*y² . b)Compute the directional derivative Daf…

A:

Q: Suppose that the vector-valued function F(t) = < 3t, cos(5t), sin(5t) represents the position of an…

A:

Q: Prove that the directional derivative of f(x, y) = y + 2xy³+x²y2 at point (0, 1) in the direction…

A: The gradient is just the vector of partial derivatives. The partial derivatives of f at the point…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.