Questions in Section Calculus III: page 4

Calculate the double integral I=iint e^x/x^2dxdy over the region shown in Fig.

Find the partial derivative f_x of the function f(x,y)=x^2e^(cos(xy)).

1. Find the tangent plane to the surface z=(4x)/y using partial derivatives of z as the function of x and y. The point where the plane touches the surface is (x,y)=(1,2).

2. Rewrite the above equation as zy-4x=0 and regard it as the expression for the level surface of the function f(x,y,z)=zy-4x. Find the same tangent plane using gradient of f.

Consider the vector v=x i+y^2j+z^3k. Show that curl(v)=0. Show that curl of any vector u=f(x)i+g(y)j+h(z)k is always zero regardless of specific form of the functions f(x),g(y) and h(z).

1. Determine whether the force F is conservative and, if it is, find its potential: F(x,y)=(ye^(xy))i+(xe^(xy)+cosy)j.

2. Determine the work done by F on moving a particle from the point (0,0) to the point (1,pi).

The Figure shows the region of integration for the double integral intint(7/2xsqrt(y)+y)dxdy.

Determine the limits of the integral and evaluate.

Calculate the double integral I=iint cos(x^2)/x^2dxdy over the region in the first quadrant restricted by the axis x, line y=x^3 and vertical line x=2. Make a sketch of the region.

Find the gradient of the function f(x,y,z)=x^2e^(cos(x/y))+ln(sin(piy)/4) at the point (pi/2,1/2,1).

Compute divergence and curl of the vector field F=(x^2-y^2)hati+4xyhatj+(x^2-xy)hatk at the point P(1,2,3).

Evaluate the integral of vector field F=x/(yz)hati+y/(xz)hatj+z/(xy)hatk along the line Gamma.

r=cost hati+sint hat j+cost hat k, where t varies from pi/6 to pi/3.

Present your result in the exact form and evaluate it up yo four decimal places.