# Calculus/Multivariable and differential calculus:Exercises

## Contents

- 1 Parametric Equations
- 2 Polar Coordinates
- 3 Calculus in Polar Coordinates
- 4 Vectors and Dot Product
- 5 Cross Product
- 6 Calculus of Vector-Valued Functions
- 7 Motion in Space
- 8 Length of Curves
- 9 Parametrization and Normal Vectors
- 10 Equations of Lines And Planes
- 11 Limits And Continuity
- 12 Partial Derivatives
- 13 Chain Rule
- 14 Tangent Planes
- 15 Maximum And Minimum Problems
- 16 Double Integrals over Rectangular Regions
- 17 Double Integrals over General Regions
- 18 Double Integrals in Polar Coordinates
- 19 Triple Integrals
- 20 Cylindrical And Spherical Coordinates
- 21 Center of Mass and Centroid
- 22 Vector Fields
- 23 Line Integrals
- 24 Conservative Vector Fields
- 25 Green's Theorem
- 26 Divergence And Curl
- 27 Surface Integrals
- 28 Stokes' Theorem
- 29 Divergence Theorem

## Parametric Equations[edit]

*P*(0,0) to

*Q*(7,17).

*x*-axis and the minor axis of length 3 along the

*y*-axis, generated clockwise.

## Polar Coordinates[edit]

*y*=

*mx*+

*b*in polar coordinates.

Sketch the following polar curves without using a computer.

Sketch the following sets of points.

## Calculus in Polar Coordinates[edit]

Find points where the following curves have vertical or horizontal tangents.

Sketch the region and find its area.

## Vectors and Dot Product[edit]

## Cross Product[edit]

Find and

Find the area of the parallelogram with sides and .

*x*-axis. Find the magnitude and the direction of the torque at the pivot when the force is applied to the wrench

*n*units away from the origin.

Prove the following identities or show them false by giving a counterexample.

## Calculus of Vector-Valued Functions[edit]

## Motion in Space[edit]

## Length of Curves[edit]

Find the length of the following curves.

## Parametrization and Normal Vectors[edit]

**T**and the principal unit normal vector

**N**for the curve Check that

**T**⋅

**N**=0.

## Equations of Lines And Planes[edit]

*x*−

*y*+

*z*=1 passing through the point (0,2,-2)

*x*+

*y*+2

*z*=4 passing through the point (5,5,5).

*x*+2

*y*−

*z*=1 and

*x*+

*y*+

*z*=1 intersect.

*x*+2

*y*−

*z*=1 and

*x*+

*y*+

*z*=1.

*x*+

*y*+

*z*=1.

## Limits And Continuity[edit]

Evaluate the following limits.

At what points is the function *f* continuous?

Use the two-path test to show that the following limits do not exist. (A path does not have to be a straight line.)

## Partial Derivatives[edit]

Find the four second partial derivatives of the following functions.

## Chain Rule[edit]

Find

Find

*x*is the side of the square base and

*h*is the height of the pyramid. Suppose that and for Find

## Tangent Planes[edit]

Find an equation of a plane tangent to the given surface at the given point(s).

## Maximum And Minimum Problems[edit]

Find critical points of the function *f*. When possible, determine whether each critical point corresponds to a local maximum, a local minimum, or a saddle point.

Find absolute maximum and minimum values of the function *f* on the set *R*.

*R*is a closed triangle with vertices (0,0), (2,0), and (0,2).

*x*−

*y*+

*z*=2 closest to the point (1,1,1).

## Double Integrals over Rectangular Regions[edit]

Evaluate the given integral over the region *R*.

Evaluate the given iterated integrals.

## Double Integrals over General Regions[edit]

Evaluate the following integrals.

*R*is bounded by

*x*=0,

*y*=2

*x*+1, and

*y*=5−2

*x*.

*R*is in the first quadrant and bounded by

*x*=0, and

Use double integrals to compute the volume of the given region.

## Double Integrals in Polar Coordinates[edit]

*R*is the unit disk centered at the origin.

## Triple Integrals[edit]

In the following exercises, sketching the region of integration may be helpful.

*x*+3

*y*+6

*z*=12 and the coordinate planes.

*y*=

*x*and

*x*=0.

*dydzdx*.

## Cylindrical And Spherical Coordinates[edit]

*z*=0 and the hyperboloid

*D*is a unit ball, use a triple integral in spherical coordinates to evaluate

## Center of Mass and Centroid[edit]

## Vector Fields[edit]

One can sketch two-dimensional vector fields by plotting vector values, flow curves, and/or equipotential curves.

## Line Integrals[edit]

*C*is the line segment from (0,0) to (5,5)

*C*is the circle of radius 4 centered at the origin

*C*is the helix

*C*is the arc of the parabola

## Conservative Vector Fields[edit]

Determine if the following vector fields are conservative on

Determine if the following vector fields are conservative on their respective domains in When possible, find the potential function.

## Green's Theorem[edit]

*y*=0 and below

*y*=

*x*(2-

*x*) in two different ways, and compare the answers.

## Divergence And Curl[edit]

## Surface Integrals[edit]

*z*=2−

*x*−

*y*in the first octant.

*z*direction.

*y*direction.

## Stokes' Theorem[edit]

where , is the upper half of the ellipsoid , and points in the direction of the

*z*-axis.

where , is the part of the sphere for , and points in the direction of the

*z*-axis.

## Divergence Theorem[edit]

Compute the net outward flux of the given field across the given surface.

*xy*-plane.