# Vector Line Integral Direction of Limits

• Master1022
In summary, the conversation discusses a disagreement on the technicality of evaluating a line integral for a given vector field. The approach involves simplifying the vector field in the given plane and setting up the integral. The book answer is deemed incorrect and the correct approach is clarified by other users. The conversation ends with an agreement on the correct solution.
Master1022
Homework Statement
Evaluate the line integral for the vector field ## v = (2x - y) \hat i + (-yz^2) \hat j + -(y^2 z) \hat k ## from (1,1) to (0,1)
Relevant Equations
Integration
Hi,

I apologise as I know I have made similar posts to this in the past and I thought I finally understood it. However, this solution seems to disagree on a technicality. I know the answer ends up as 0, but I still want to understand this from a conceptual point.

Question: Evaluate the line integral for the vector field ## v = (2x - y) \hat i + (-yz^2) \hat j + -(y^2 z) \hat k ## from (1,1) to (0,1) when we are in the xy-plane (i.e. z = 0).

Approach:
Given that we are in the xy-plane where ## z = 0 ## and at ## y = 1 ##, the vector field becomes:
$$v = (2x - 1) \hat i$$

Then when I set up the integral, ## d \vec r = -dx \hat i ## and thus:
$$\int_0^1 \vec v \cdot d \vec r = \int_0^1 (2x - 1) \hat i \cdot -dx \hat i = - \int_0^1 (2x - 1) dx = 0$$

However, the answer writes the integral as:
$$- \int_1^0 (2x - 1) dx = 0$$

I know that the answers are the same, but if the integral wasn't 0, then the answers would be different. I thought the convention was to define ## d \vec r ## in the direction of the path and the limits in terms of the increasing parameter.

I am sure this has been answered in similar posts but am unable to find them. Any guidance would be greatly appreciated.

Thanks.

Seems to me the book answer is wrong ...
(@haruspex ?)

Master1022
BvU said:
Seems to me the book answer is wrong ...
(@haruspex ?)
I agree.
(Odd to write the range with endpoints expressed as only 2D.)

BvU and Master1022

## 1. What is a vector line integral?

A vector line integral is a mathematical concept used in vector calculus to calculate the total effect of a vector field along a given path or curve. It takes into account both the magnitude and direction of the vector field at each point along the path.

## 2. What is the direction of a vector line integral?

The direction of a vector line integral is determined by the direction of the vector field at each point along the path. It can be either positive or negative, depending on whether the vector field is aligned or opposed to the direction of the path.

## 3. How is the direction of a vector line integral calculated?

The direction of a vector line integral is calculated by taking the dot product of the vector field and the tangent vector of the path at each point. The resulting value is then integrated along the path to determine the total effect.

## 4. What are the limits of a vector line integral?

The limits of a vector line integral refer to the starting and ending points of the path along which the integral is being calculated. These limits can be defined by specific coordinates or by a parametric equation.

## 5. What is the significance of the direction of limits in a vector line integral?

The direction of limits in a vector line integral is important because it determines the direction in which the integral is being calculated. This can affect the final value of the integral and its interpretation in the context of the vector field.

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