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.

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